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CBSE Questions for Class 11 Commerce Applied Mathematics Tangents And Its Equations Quiz 6 - MCQExams.com
CBSE
Class 11 Commerce Applied Mathematics
Tangents And Its Equations
Quiz 6
The points of contact of the vertical tangents
x
=
2
−
3
sin
θ
,
y
=
3
+
2
cos
θ
are
Report Question
0%
(
2
,
5
)
,
(
2
,
1
)
0%
(
−
1
,
3
)
,
(
5
,
3
)
0%
(
2
,
5
)
,
(
5
,
3
)
0%
(
−
1
,
3
)
,
(
2
,
1
)
Explanation
For the tangents to be vertical,
d
y
d
x
=
∞
Or
d
x
d
y
=
0
Or
d
x
d
θ
=
0
Or
−
3
cos
θ
=
0
Or
θ
=
2
n
−
1
2
π
Hence
θ
=
π
2
,
3
π
2
.
Now
x
π
2
=
−
1
y
π
2
=
3
.
Similarly
x
3
π
2
=
5
y
3
π
2
=
3
.
Hence the points are
(
−
1
,
3
)
and
(
3
,
5
)
.
Tangent is drawn to ellipse
x
2
27
+
y
2
=
1
at
(
3
√
3
cos
θ
,
sin
θ
)
(where
θ
∈
(
0
,
π
2
)
)
.
Then the value of
θ
such that sum of intercepts on axes made by this tangent is least is
Report Question
0%
π
3
0%
π
6
0%
π
8
0%
π
4
Explanation
Let
x
=
3
√
3
cos
θ
And
y
=
sin
θ
Hence,
d
y
d
x
=
−
cos
θ
3
√
3
sin
θ
...(i)
Hence equation of the tangent will be
y
−
sin
θ
=
−
cos
θ
3
√
3
sin
θ
(
x
−
3
√
3
.
cos
θ
)
3
√
3
sin
θ
.
y
−
3
√
3
sin
2
θ
=
−
x
cos
θ
+
3
√
3
cos
2
θ
3
√
3
sin
θ
.
y
+
x
cos
θ
=
3
√
3
.
Hence the intercepts are
c
o
s
e
c
θ
and
3
√
3
.
sec
θ
.
Hence,
k
=
c
o
s
e
c
θ
+
3
√
3
.
sec
θ
Now
d
k
d
θ
=
−
c
o
s
e
c
θ
.
cot
θ
+
3
√
3
.
sec
θ
.
tan
θ
=
0
3
√
3
.
sec
θ
.
tan
θ
=
c
o
s
e
c
θ
.
cot
θ
3
√
3
.
tan
2
θ
.
1
cos
θ
=
1
sin
θ
3
√
3
.
tan
2
θ
=
cos
θ
sin
θ
3
√
3
.
tan
3
θ
=
1
tan
3
θ
=
1
3
√
3
tan
θ
=
1
√
3
θ
=
tan
−
1
(
1
√
3
)
θ
=
30
0
=
π
6
.
The tangent to the curve
3
x
y
2
−
2
x
2
y
=
1
at
(
1
,
1
)
meets the curve again at the point
Report Question
0%
(
16
5
,
1
20
)
0%
(
−
16
5
,
−
1
20
)
0%
(
1
20
,
16
5
)
0%
(
−
1
20
,
16
5
)
Explanation
Given,
3
x
y
2
−
2
x
2
y
=
1
Differentiating with respect to x, gives us
3
y
2
+
6
x
y
y
′
−
4
x
y
−
2
x
2
y
′
=
0
At (1,1)
3
+
6
y
′
−
4
−
2
y
′
=
0
4
y
′
=
1
y
′
=
1
4
Hence the equation of the tangent is
y
−
1
x
−
1
=
1
4
4
y
−
4
=
x
−
1
x
−
4
y
=
−
3
x
=
4
y
−
3
Substituting in the equation of the curve, give us
3
(
4
y
−
3
)
y
2
−
2
(
4
y
−
3
)
2
y
=
1
(
4
y
−
3
)
y
[
3
y
−
2
(
4
y
−
3
)
]
=
1
(
4
y
−
3
)
y
(
6
−
5
y
)
=
1
y
(
4
y
−
3
)
(
5
y
−
6
)
=
−
1
y
(
20
y
2
−
24
y
−
15
y
+
18
)
=
−
1
y
(
20
y
2
−
39
y
+
18
)
=
−
1
20
y
3
−
39
y
2
+
18
y
+
1
=
0
y
=
1
,
−
1
20
⇒
x
=
1
,
−
16
5
Hence the tangent again cuts the curve at
(
−
16
5
,
−
1
20
)
The angle at which the curve
y
=
k
e
k
x
intersects the
y
-axis is
Report Question
0%
tan
−
1
(
k
2
)
0%
cot
−
1
(
k
2
)
0%
sin
−
1
(
1
/
√
1
+
k
4
)
0%
sec
−
1
(
1
/
√
1
+
k
4
)
Explanation
d
y
d
x
=
k
2
e
k
x
.
The curve intersects
y
-axis at
(
0
,
k
)
So,
d
y
d
x
|
(
0
,
k
)
=
k
2
.
If
θ
is the angle at which the given
curve intersects the
y
-axis then
tan
(
π
/
2
−
θ
)
=
k
2
−
0
1
+
0.
k
2
=
k
2
.
Hence
θ
=
cot
−
1
k
2
The lines tangent to the curves
y
3
−
x
2
y
+
5
y
−
2
x
=
0
and
x
4
−
x
3
y
2
+
5
x
+
2
y
=
0
at the origin intersect at an angle
θ
equal to
Report Question
0%
π
6
0%
π
4
0%
π
3
0%
π
2
Let
f
(
x
)
=
x
3
+
a
x
+
b
with
a
≠
b
and suppose the tangent lines to the graph of
f
at
x
=
a
and
x
=
b
have the same gradient Then the value of
f
(
1
)
is equal to
Report Question
0%
0
0%
1
0%
−
1
3
0%
2
3
Explanation
f
(
x
)
=
x
3
+
a
x
+
b
f
′
(
x
)
=
3
x
2
+
a
Given gradient at
x
=
a
and at
x
=
b
are same
⇒
3
a
2
+
a
=
3
b
2
+
a
⇒
b
2
=
a
2
But given
a
≠
b
⇒
a
+
b
=
0
.
.
(
1
)
Hence
f
(
1
)
=
1
+
a
+
b
=
1
using (1)
A curve with equation of the form
y
=
a
x
4
+
b
x
3
+
c
x
+
d
has zero gradient at the point (0, 1) and also touches the x-axis at the point (-1, 0) then the values of x for which the curve has a negative gradient are
Report Question
0%
x > -1
0%
x < 1
0%
x < -1
0%
−
1
≤
×
≤
1
Explanation
Given,
y
=
a
x
4
+
b
x
3
+
c
x
+
d
⇒
y
′
=
4
a
x
3
+
3
b
x
2
+
c
Using given conditions,
y
(
0
)
=
1
⇒
d
=
1
y
′
(
0
)
=
0
⇒
c
=
0
y
(
−
1
)
=
0
⇒
a
−
b
=
−
1
.
.
(
1
)
and
y
′
(
−
1
)
=
0
⇒
4
a
−
3
b
=
0
.
.
(
2
)
Solving equation (1) and (2) we get,
a
=
3
,
b
=
4
Hence the polynomial is,
y
=
3
x
4
+
4
x
3
+
1
y
′
=
12
x
2
(
1
+
x
)
Now for negative gradient
y
′
<
0
⇒
12
x
2
(
1
+
x
)
<
0
⇒
x
<
−
1
For the curve represented parametrically by the equations
x
=
2
ln
cot
t
+
1
and
y
=
tan
t
+
cot
t
Report Question
0%
normal at
t
=
π
4
is parallel to y=axis
0%
tangent at
t
=
π
4
is parallel to x-axis
0%
tangent at
t
=
π
4
is parallel to the line
y
=
x
0%
normal at
t
=
π
4
is parallel to the line
y
=
x
Explanation
Given,
x
=
2
ln
cot
t
+
1
and
y
=
tan
t
+
cot
t
∴
d
x
d
t
=
2
(
−
c
o
s
e
c
2
t
)
cot
t
at
t
=
π
4
,
d
x
d
t
=
−
4
and
d
y
d
t
=
sec
2
t
−
c
o
s
e
c
2
t
a
t
t
=
π
4
d
y
d
t
=
0
d
y
d
x
=
0
for tangent & hence it is parallel to x-axis & its normal is parallel to y axis
If a variable tangent to the curve
x
2
y
=
c
3
makes intercepts
a
,
b
on
x
and
y
axis respectively then the value of
x
2
is
Report Question
0%
27
c
3
0%
4
27
c
3
0%
27
4
c
3
0%
4
9
c
3
Explanation
Let any point on the curve is
(
c
t
,
c
t
2
)
y
=
c
3
x
2
⇒
d
y
d
x
=
−
2
c
3
x
3
=
−
2
c
3
c
3
t
3
d
y
d
x
=
−
2
t
3
Equation of tangent is
y
−
c
t
2
=
−
2
t
3
(
x
−
c
t
)
for x intercept
0
−
c
t
2
=
−
2
t
3
(
a
−
c
t
)
⇒
c
2
t
=
a
−
c
t
a
=
3
2
c
t
for y intercept
b
−
c
t
2
=
−
2
t
3
(
0
−
c
t
)
⇒
b
−
c
t
2
=
2
t
2
c
⇒
b
=
3
c
t
2
a
2
b
=
9
4
,
c
2
t
2
×
3
c
t
2
=
27
c
3
4
The coordinates of the point(s) on the graph of the function
f
(
x
)
=
x
3
3
−
5
x
2
2
+
7
x
−
4
where the tangent drawn cut off intercepts from the coordinate axes which are equal in magnitude but opposite in sign is
Report Question
0%
(
2
,
8
3
)
0%
(
3
,
7
2
)
0%
(
1
,
5
6
)
0%
none
Explanation
Hence
d
y
d
x
=
1
at that point.
Hence
x
2
−
5
x
+
7
=
1
Or
x
2
−
5
x
+
6
=
0
Or
(
x
−
2
)
(
x
−
3
)
=
0
Or
x
=
2
and
x
=
3
f
(
2
)
=
8
3
And
f
(
3
)
=
7
2
Hence the points are
(
2
,
8
3
)
and
(
3
,
7
2
)
.
The number of values of c such that the straight line
3
x
+
4
y
=
c
touches the curve
x
4
2
=
x
+
y
is
Report Question
0%
0
0%
1
0%
2
0%
4
Explanation
3
x
+
4
y
=
c
Or
3
+
4
y
′
=
0
Or
y
′
=
−
3
4
.
This is the slope of the tangent.
Now
Differentiating the equation of the curve with respect to
x
, we get
2
x
3
=
1
+
y
′
Now
y
′
=
−
3
4
Hence
2
x
3
=
1
4
Or
x
3
=
1
8
Hence
x
=
0.5
Substituting in the equation of the curve we get
1
32
=
1
2
+
y
Or
y
=
−
15
32
.
Hence substituting the above points on the equation of the line, we get
3
2
+
−
15
8
=
c
Or
c
=
−
3
8
Hence there is only
1
value of
c
.
Find all the tangents to the curve
y
=
cos
(
x
+
y
)
,
−
2
π
≤
×
≤
2
π
that are parallel to the line
x
+
2
y
=
0
Report Question
0%
x
+
2
y
=
π
2
&
x
+
2
y
=
−
3
π
2
0%
x
+
2
y
=
π
2
&
x
+
2
y
=
−
π
2
0%
x
+
2
y
=
−
3
π
2
&
x
+
2
y
=
−
3
π
2
0%
x
+
2
y
=
3
π
2
&
x
+
2
y
=
−
3
π
2
Explanation
Given equation is
x
+
2
y
=
0
Slope
=
−
1
2
=
−
1
2
=
−
s
i
n
(
x
+
y
)
(
1
+
y
′
)
⇒
−
1
2
=
−
sin
(
x
+
y
)
(
1
−
1
2
)
⇒
sin
(
x
+
y
)
=
1
⇒
cos
(
x
+
y
)
=
0
⇒
y
=
0
cos
x
=
0
∴
x
=
π
2
,
3
π
2
,
−
π
2
.
−
3
π
2
sin
x
=
1
is possible for x =
π
2
or
−
3
π
2
Equation are :
y
−
0
=
−
1
2
(
x
−
π
2
)
and
y
−
0
=
−
1
2
(
x
+
3
π
2
)
The angle at which the curve
y
=
k
e
k
x
intersects the y - axis is
Report Question
0%
tan
−
1
k
2
0%
cot
−
1
(
k
2
)
0%
sin
−
1
(
1
√
1
+
k
4
)
0%
sec
−
1
(
√
1
+
k
4
)
Explanation
d
y
d
x
=
K
2
e
k
x
d
y
d
x
|
x
=
0
=
K
2
=
tan
θ
(where
ϕ
is angle made by x-axis)
Let
ϕ
be the gngle made by y-axis
tan
θ
=
tan
(
3
π
2
−
ϕ
)
=
cot
ϕ
cot
ϕ
=
K
2
ϕ
=
cot
−
1
(
K
2
)
⇒
ϕ
=
sin
−
1
(
1
√
1
+
K
4
)
The abscissa of the point on the curve
√
x
y
=
a
+
x
the tangent at which cuts off equal intercepts from the co-ordinate axes is (a > 0)
Report Question
0%
a
√
2
0%
−
a
√
2
0%
a
√
2
0%
−
a
√
2
Explanation
y
=
(
a
+
x
)
2
x
⇒
y
=
a
2
x
+
2
a
+
x
d
y
d
x
=
−
a
2
x
2
+
1
=
−
1
( for equal intercepts)
x
2
=
a
2
2
⇒
x
=
±
a
√
2
Consider the curve
f
(
x
)
=
x
1
/
3
then
Report Question
0%
the equation of tangent at (0, 0) is x = 0
0%
the equation of normal at (0, 0) is y = 0
0%
normal to the curve does not exist at (0, 0)
0%
f(x) and its inverse meet at exactly 3 points
Explanation
f
(
x
)
=
x
1
/
3
⇒
f
′
(
x
)
=
1
3
x
2
/
3
f
′
(
0
)
→
∞
tangent is vertical at
x
=
0
Equation of tangent at
(
0
,
0
)
is
x
=
0
Equation of normal is
y
=
0
f
(
x
)
=
f
−
1
(
x
)
⟹
x
1
3
=
x
3
⟹
x
9
=
x
⟹
x
=
0
,
1
,
−
1
Equation of the line through the point
(
1
/
2
,
2
)
and tangent to the parabola
y
=
−
x
2
2
+
2
and secant to the curve
y
=
√
4
−
x
2
is
Report Question
0%
2
x
+
2
y
−
5
=
0
0%
2
x
+
2
y
−
9
=
0
0%
y
−
2
=
0
0%
none
Explanation
Equation of tangent is
y
−
2
=
m
(
x
−
1
2
)
y
=
m
x
+
2
−
m
2
Put it in the parabolas
m
x
+
2
−
m
2
=
−
x
2
2
+
2
x
2
2
+
m
x
−
m
2
=
0
since D = 0
⇒
m
2
+
m
=
0
m = 0, -1 Two tangents are there
(i) y = 2
(ii)
y
=
−
x
+
2
+
1
2
⇒
y
=
−
x
+
5
2
y
−
=
−
x
+
5
2
The line y = 2 is tangent but
y
=
−
x
+
5
2
is secant for the curve
Equation of a trangent to the curve
y
cot
x
=
y
3
tan
x
at the point where the abscissa is
π
4
is
Report Question
0%
4
x
+
2
y
=
π
+
2
0%
4
x
−
2
y
=
π
+
2
0%
x = 0
0%
y = 0
Explanation
Put
x
=
π
4
in the given equation,
y
=
y
3
⇒
y
(
y
2
−
1
)
=
0
⇒
y
=
−
1
,
0
,
1
So the points are
(
π
4
,
−
1
)
,
(
π
4
,
0
)
and
(
π
4
,
1
)
Now given equation may be written as,
y
=
tan
2
x
.
y
3
Differentiating w.r.t
x
d
y
d
x
=
tan
2
x
.3
y
2
d
y
d
x
+
y
3
.2
tan
x
.
sec
2
x
⇒
d
y
d
x
=
2
y
3
sec
2
x
.
tan
x
1
−
3
tan
2
x
.
y
2
Thus slope of tangents are,
m
1
=
(
d
y
d
x
)
(
π
4
,
−
1
)
=
2
(
−
1
)
.2
.1
1
−
3.1.1
=
2
m
2
=
(
d
y
d
x
)
(
π
4
,
0
)
=
2
(
0
)
.2
.1
1
−
3.1.0
=
0
m
3
=
(
d
y
d
x
)
(
π
4
,
1
)
=
2
(
1
)
.2
.1
1
−
3.1.1
=
−
2
Hence equation of tangents are,
(
y
+
1
)
=
2
(
x
−
π
4
)
⇒
4
x
−
2
y
=
π
+
2
(
y
−
0
)
=
0
(
x
−
π
4
)
⇒
y
=
0
(
y
−
1
)
=
2
(
x
−
π
4
)
⇒
4
x
+
2
y
=
π
+
2
If the curve
(
x
a
)
n
+
(
y
b
)
n
=
2
touches the straight line
x
a
+
y
b
=
2
, then find the value of
n
.
Report Question
0%
2
0%
3
0%
4
0%
any real number
Explanation
Given
(
x
a
)
n
+
(
y
b
)
n
=
2
Differentiating both sides w.r.t
x
,
we get
n
a
(
x
a
)
n
−
1
+
b
b
(
y
b
)
n
−
1
×
d
y
d
x
=
0
⇒
d
y
d
x
=
−
n
a
(
x
a
)
n
−
1
×
b
a
(
b
y
)
n
−
1
∴
d
y
d
x
at
(
a
,
b
)
=
b
a
∴
Tangent is
y
−
b
=
−
b
a
(
x
−
a
)
⇒
b
x
+
a
y
=
2
a
b
⇒
x
a
+
y
b
=
2
for all values of
n
(
∵
d
y
d
x
is independent of
n
)
Find the equation of the normal to the curve
y
=
(
1
+
x
)
y
+
sin
−
1
(
sin
2
x
)
at
x
=
0
Report Question
0%
x
+
y
−
1
=
0
0%
x
+
y
−
2
=
0
0%
x
+
y
−
3
=
0
0%
x
+
y
−
4
=
0
Explanation
Clearly at
x
=
0
,
y
=
1
Thus curve is,
y
=
(
1
+
x
)
y
+
sin
−
1
(
sin
2
x
)
Differentiating w.r.t
x
d
y
d
x
=
(
1
+
x
)
y
[
y
1
+
x
+
log
(
1
+
x
)
.
d
y
d
x
]
+
2
sin
x
.
cos
x
√
1
+
sin
4
x
Putting
x
=
0
,
y
=
1
, we get,
d
y
d
x
=
1
=
m
(say)
Thus slope of the normal is
=
−
1
m
=
−
1
Hence require equation is given by,
(
y
−
1
)
=
−
1
(
x
−
0
)
⇒
x
+
y
=
1
Find the equation of normal to the curve
x
2
=
4
y
passing through the point
(
1
,
2
)
Report Question
0%
x
+
y
=
3
0%
x
−
y
=
3
0%
2
x
−
y
=
4
0%
2
x
−
3
y
=
1
Explanation
x
2
=
4
y
⇒
d
y
d
x
=
x
2
=
m
(say)
Thus slope of normal at
(
1
,
2
)
is
m
′
=
−
1
m
=
−
2
x
=
−
2
2
=
−
1
Hence, equation of normal at
(
1
,
2
)
to the parabola is,
(
y
−
2
)
=
−
1
(
x
−
1
)
⇒
x
+
y
=
3
A function is defined parametrically by the equations
x=
2
t
+
t
2
sin
1
t
if
t
≠
0
;
0
, otherwise
and
y =
1
t
sin
t
2
if
t
≠
0
;
0
, otherwise
Find the equation of the tangent and normal at the point for t = 0 if they exist
Report Question
0%
Tangent ;
2
y
−
x
=
0
;
Normal:
2
x
+
y
=
0
0%
Tangent ;
3
y
−
x
=
0
;
Normal:
2
x
+
y
=
0
0%
Tangent ;
2
y
−
x
=
0
;
Normal:
3
x
+
y
=
0
0%
Tangent ;
3
y
−
x
=
0
;
Normal:
3
x
+
y
=
0
Explanation
Given,
x
=
2
t
+
t
2
sin
1
t
and
y
=
1
t
sin
t
2
At
t
=
0
the point is origin
d
x
d
t
=
lim
\displaystyle \frac{dy}{dt}=\lim_{t-0}\frac{\dfrac{1}{t}\sin t^{2}}{t}=1
\displaystyle \frac{dy}{dx}=\dfrac{1}{2}
equation of tangent is
\displaystyle y-0=\frac{1}{2}\left ( x-0 \right )\Rightarrow 2y-x=0
equation of normal is
y - 0 = -2(x - 0)\Rightarrow y+2x=0
The curve
\displaystyle y=ax^{3}+bx^{2}+cx+5
touches the
x
- axis at
P(-2, 0)
and cuts the
y
-axis at a point
Q
, where its gradient is
3
. Find
a, b, c
.
Report Question
0%
\displaystyle a=-\frac{1}{5}, b=1,c=3
0%
\displaystyle a=-\frac{1}{4}, b=-1,c=4
0%
\displaystyle a=-\frac{1}{4}, b=0,c=3
0%
\displaystyle a=-\frac{1}{3}, b=1,c=-3
Explanation
Given,
y= ax^3+bx^2+cx+5
\therefore \displaystyle \frac{dy}{dx}= 3ax^{2}+2bx+c
Since the curve touches
x
-axis at
\left ( -2,0 \right )
so
\displaystyle \frac{dy}{dx}|_{\left ( -2,0 \right )}= 0\Rightarrow 12a-4b+c= 0
....
\left ( i \right )
The curve cut the
y
-axis at
\left ( 0,8 \right )
, so
\displaystyle \frac{dy}{dx}|_{\left ( 0,8 \right )}= 3\Rightarrow c= 3
Also the curve passes through
\left ( -2,0 \right )
, so
0= -8a+4b-2c+8\Rightarrow -8a+4b-2= 0
.....
\left ( ii \right )
Solving
\left ( i \right )
and
\left ( ii \right )
a = -\dfrac {1}{4}
,
b =0
The slope of the normal to the curve
\displaystyle x=a\left ( \theta -\sin \theta \right ),\: \: y=a\left ( 1-\cos \theta \right )
at point
\displaystyle \theta =\dfrac{\pi }2
is
Report Question
0%
0
0%
1
0%
-1
0%
\dfrac 1{\displaystyle \sqrt{2}}
Explanation
The slope of normal
\displaystyle x=a\left ( \theta -\sin \theta \right );\: \: \: \: y=a\left ( 1-\cos \theta \right )\: \: \: at\: \: \: \theta =\dfrac{\pi }{2}
\displaystyle \left ( \dfrac{dx}{d\theta } \right )_{\theta =\dfrac{\pi }{2}}=a\left ( 1-\cos \theta \right )=a
\displaystyle \left ( \dfrac{dy}{d\theta } \right )_{\theta =\dfrac{\pi }{2}}=a\sin \theta =a
\therefore \left ( \dfrac{dy}{dx} \right )=1
Therefore, slope of normal at the given point is,
m= \left ( -\dfrac{dx}{dy} \right )=-1
Hence, option 'C' is correct.
The equation of the normal to the curve
\displaystyle y^{2}=4ax
at point
(a, 2a)
is
Report Question
0%
x - y + a = 0
0%
x + y - 3a = 0
0%
x + 2y + 4a = 0
0%
x + y + 4a = 0
Explanation
\displaystyle y^{2}=4ax
\displaystyle 2y\frac{dy}{dx}=4a\Rightarrow \cfrac{dy}{dx}=\cfrac{2a}{y}
\therefore \displaystyle \left ( \frac{dy}{dx} \right )_{\left ( a,2a \right )}=\left ( \frac{2a}{y} \right )_{\left ( a,2a \right )}=1 =m
(say)
\displaystyle \therefore
Slope of normal at the given point is
=-\cfrac{1}{m}=-1
Therefore, the equation of normal is,
(y - 2a) = -1 (x - a)
\Rightarrow x + y = 3a
Hence, option 'B' is correct.
The tangent at a point
P
of a curve meets the
y-
axis at
A
and the line parallel to
y-
axis at
A
, and the line parallel to
y-
axis through
P
meets the
x-
axis at
B
. If area of
\Delta OAB
is constant (
O
being the origin). Then the curve is
Report Question
0%
cx^2-xy+k=0
0%
x^{2}+y^{2}=cx
0%
3x^{2}+4y^{2}=k
0%
xy-x^{2}y^{2}+kx=0
A tangent to the hyperbola
\displaystyle y=\frac { x+9 }{ x+5 }
passing though the origin is
Report Question
0%
x+25y=0
0%
5x+y=0
0%
5x-y=0
0%
x-25y=0
Explanation
\displaystyle y=\frac { x+9 }{ x+5 } =1+\frac { 4 }{ x+5 }
\displaystyle \frac { dy }{ dx }
at
\displaystyle \left( { x }_{ 1 },{ y }_{ 1 } \right) =-\frac { 4 }{ { \left( { x }_{ 1 }+5 \right) }^{ 2 } }
Equation of tangent is
\displaystyle y-{ y }_{ 1 }=-\frac { 4 }{ { \left( { x }_{ 1 }+5 \right) }^{ 2 } } \left( x-{ x }_{ 1 } \right) \Rightarrow y-1-\frac { 4 }{ { x }_{ 1 }+5 } =-\frac { 4 }{ { \left( { x }_{ 1 }+5 \right) }^{ 2 } } \left( x-{ x }_{ 1 } \right)
Since it passes through origin
(0,0)
\displaystyle -1-\frac { 4 }{ { x }_{ 1 }+5 } =-\frac { 4 }{ { \left( { x }_{ 1 }+5 \right) }^{ 2 } } \Rightarrow { \left( { x }_{ 1 }+5 \right) }^{ 2 }+4\left( { x }_{ 1 }+5 \right) +4{ x }_{ 1 }=0
\Rightarrow { { x }_{ 1 } }^{ 2 }+18{ x }_{ 1 }+45=0\Rightarrow \left( { x }_{ 1 }+15 \right) \left( { x }_{ 1 }+3 \right) =0\Rightarrow { x }_{ 1 }=15
or
{ x }_{ 1 }=-3
So equation of tangent is
\displaystyle y-1-\frac { 4 }{ \left( -15+5 \right) } =-\frac { 4 }{ { \left( -15+5 \right) }^{ 2 } } \left( x+15 \right)
\displaystyle \Rightarrow y-1+\frac { 2 }{ 5 } =-\frac { 1 }{ 25 } \left( x+15 \right) \Rightarrow y-\frac { 3 }{ 5 } =-\frac { x }{ 25 } -\frac { 3 }{ 5 } \Rightarrow x+25y=0
asymptotes of the graph
Report Question
0%
\displaystyle x=\frac{3\pi }{2}
0%
\displaystyle x=-\frac{\pi }{2}
0%
\displaystyle x=\frac{\pi }{2}
0%
\displaystyle x=-\frac{3\pi }{2}
Explanation
f'(x)=\dfrac{\cos(x)}{1+\sin(x)}
Or
\dfrac{dy}{dx}=\dfrac{\cos x}{1+\sin x}
For asymptotes,
\dfrac{dy}{dx}=\infty
Or
dx=0
Or
\sin(x)=-1
Or
x=\dfrac{3\pi}{2},\dfrac{-\pi}{2}
.
Let
f
be a continuous, differentiable and bijective function. If the tangent to
y=f\left( x \right)
at
x=b
, then there exists at least one
c\in \left( a,b \right)
such that
Report Question
0%
f'\left( c \right) =0
0%
f'\left( c \right) >0
0%
f'\left( c \right) <0
0%
none of these
Explanation
Since the same line is tangent at one point
x=a
and normal at other point
x=b
\Rightarrow
Tangent at
x=b
will be perpendicular to tangent at
x=a
\Rightarrow
Slope of tangent changes from positive to negative or negative to positive. Therefore, it takes the value zero somewhere. Thus, there exists a point
c\in \left( a,b \right)
where
f'\left( c \right) =0
.
The slope of normal to the curve
\displaystyle y^{2}=4ax
at a point
\displaystyle \left ( at^{2},2at \right )
is
Report Question
0%
\dfrac 1 t
0%
t
0%
-t
0%
-\dfrac 1 t
Explanation
y^2=4ax
\Rightarrow 2y\cfrac{dy}{dx}=4a\Rightarrow \cfrac{dy}{dx}=\cfrac{2a}{y}
Therefore, slope of normal to the given curve is,
=\left(-\cfrac{dx}{dy}\right)_{(at^2,2at)}=-\cfrac{2at}{2a}=-t
Hence, option 'C' is correct.
The coordinates of the point P on the graph of the function
\displaystyle y=e^-{\left | x \right |},
where area of triangle made by tangent and the coordinate axis has the greatest area, is
Report Question
0%
\displaystyle \left ( 1,\frac{1}{e} \right )
0%
\displaystyle \left ( -1,\frac{1}{e} \right )
0%
\displaystyle \left ( e,e^{-e} \right )
0%
none
Explanation
The curve of f(x) is symmetrical about y axis.
Let the point of contact of the tangent be
(a,e^{-a})
Therefore
\dfrac{dy}{dx}_{(a,e^{-a})}=-e^{-a}
Hence the equation of the tangent will be
y-e^{-a}=-e^{-a}(x-a)
y-e^{-a}=-e^{-a}x+ae^{-a}
y+e^{-a}x=e^{-a}(1+a)
\dfrac{y}{e^{-a}(1+a)}+\dfrac{x}{1+a}=1
Hence the area formed by the tangent and the coordinate axes will be
=\dfrac{(x-intercept)(y-intercept)}{2}
A=\dfrac{e^{-a}(1+a)^{2}}{2}
Differentiating with respect to a, we get
\dfrac{dA}{da}=\dfrac{1}{2}[2(1+a)e^{-a}-(1+a)^{2}e^{-a}]
=\dfrac{(1+a)e^{-a}}{2}[2-(1+a)]
=\dfrac{(1+a)e^{-a}}{2}(1-a)
=0
Hence
a=\pm1
Thus we get the points as
(1,e^{-1})
and
(-1,e^{-1})
.
On the ellipse,
4x^2\, +\, 9y^2\, =\, 1
, the points at which the tangents are parallel to the line
8x = 9y
are
Report Question
0%
\left ( \displaystyle \frac{2}{5},\,\frac{1}{5} \right )
0%
\left ( -\displaystyle \frac{2}{5},\,\frac{1}{5} \right )
0%
\left ( -\displaystyle \frac{2}{5},\,-\frac{1}{5} \right )
0%
\left ( \displaystyle \frac{2}{5},\,-\frac{1}{5} \right )
Explanation
We have
{ 4x }^{ 2 }+9{ y }^{ 2 }=1
...(1)
Differentiating w.r.t x, we get
\displaystyle 8x+18y\frac { dy }{ dx } =0\Rightarrow \frac { dy }{ dx } =-\frac { 4x }{ 9y }
The tangent at point
\left( x,y \right)
will be parallel to the line
8x=9y
if
\displaystyle \frac { -4x }{ 9y } =\frac { 8 }{ 9 } \Rightarrow x=-2y
Substituting this in (1), we get
\displaystyle 4{ \left( -2y \right) }^{ 2 }+{ 9y }^{ 2 }=1\Rightarrow { 25y }^{ 2 }=1\Rightarrow y=\pm \frac { 1 }{ 5 }
Thus the point where the tangent are parallel to
8x=9y
are
\displaystyle \left( \frac { -2 }{ 5 } ,\frac { 1 }{ 5 } \right)
and
\displaystyle \left( \frac { 2 }{ 5 } ,\frac { -1 }{ 5 } \right)
The slope of the tangent to the curve
xy + ax - by = 0
at the point
(1, 1)
is
2
then values of
a
and
b
are respectively -
Report Question
0%
1, 2
0%
2, 1
0%
3, 5
0%
None of these
Explanation
Given curve is
xy + ax - by = 0
Slope of tangent at
(1, 1)
is
\displaystyle x.\frac{dy}{dx}+y+a-b.\frac{dy}{dx}=0
\displaystyle \left ( \frac{dy}{dx} \right )_{\left ( 1,1 \right )}=-\left [ \frac{\left ( a+y \right )}{x-b} \right ]_{\left ( 1,1 \right )}=-\frac{\left ( a+1 \right )}{1-b}
\displaystyle \therefore -\frac{\left ( a+1 \right )}{1-b}=2
So,
2b - a = 3
..........(1)
\displaystyle \because (1, 1)
lies on curve
xy + ax - by = 0
\displaystyle \therefore a-b=-1
.........(2)
From (1) and (2), we get
a = 1, b = 2
Hence, option 'A' is correct.
It
\displaystyle x=t^{2}
and
y = 2t
then equation of normal at
t = 1
is -
Report Question
0%
x + y + 3 = 0
0%
x + y + 1 = 0
0%
x + y - 1 = 0
0%
x + y - 3 = 0
Explanation
\displaystyle x=t^{2},\: \: y=2t
\Rightarrow \displaystyle \frac{dx}{dt}=2t\: \: \: \: \: \:, \frac{dy}{dt}=2
\Rightarrow \displaystyle \frac{dy}{dx}=\frac{1}{t}
\therefore
Slope of normal is
=\displaystyle \left ( \frac{-dx}{dy} \right )_{t=1}=-t=-1
Therefore, equation of normal is,
(y - 2) = -1(x - 1)
\Rightarrow x + y = 3
Hence, option 'D' is correct.
The equation of the tangent to the curve
\displaystyle \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}=\frac{2}{\sqrt{a}}
at point
(a, a)
is
Report Question
0%
\displaystyle \frac{a}{\sqrt{x}}+\frac{a}{\sqrt{y}}={2}\sqrt a
0%
x + y = 2a
0%
\displaystyle \sqrt{x}+\sqrt{y}=2\sqrt{a}
0%
None of these
Explanation
Given curve is,
\displaystyle \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}=\frac{2}{\sqrt{a}}
\Rightarrow x^{-1/2}+y^{-1/2}=2a^{-1/2}
Differentiating w.r.t
x
\dfrac{-1}{2}x^{-3/2}-\dfrac{1}{2}y^{-3/2}\cfrac{dy}{dx}=0
\Rightarrow \cfrac{dy}{dx}=-\cfrac{y\sqrt{y}}{x\sqrt{x}}
Thus slope of tangent at
(a,a)
is
m=-1
The equation of required tangent is given by,
(y-a)=-1(x-a)\Rightarrow x+y=2a
Hence, option 'B' is correct.
Consider the curved mirror
y=f(x)
passing through
(0, 6)
having the property that all light rays emerging from origin, after getting reflected from the mirror becomes parallel to
x-
axis, then the equation of curve is
Report Question
0%
y^2=4(x-y)
or
y^2=36(9+x)
0%
y^2=4(1-x)
or
y^2=36(9-x)
0%
y^2=4(1+x)
or
y^2=36(9-x)
0%
None\ of\ these
The point where the tangent line to the curve
\displaystyle y=e^{2x}
at
(0, 1)
meets
x
- axis is -
Report Question
0%
(1, 0)
0%
(-1, 0)
0%
\displaystyle \left ( -\dfrac12,0 \right )
0%
None of these
Explanation
\displaystyle y=e^{2x}
\displaystyle \frac{dy}{dx}=e^{2x}\times 2
\therefore \displaystyle \left ( \frac{dy}{dx} \right )_{\left ( 0,1 \right )}=2
Thus equation of tangent at point
(0,1)
is,
y - 1 = 2(x - 0)\Rightarrow y=2x+1
Now substitute
y = 0
to get point of intersection of tangent with
x-axis
\displaystyle x=-\frac{1}{2}
Therefore, required point is
\displaystyle \left ( -\frac{1}{2},0 \right )
Hence, option 'C' is correct.
The slope of the tangents to the curve
y = (x + 1) (x - 3)
at the points where it crosses x - axis are
Report Question
0%
\displaystyle \pm 2
0%
\displaystyle \pm 3
0%
\displaystyle \pm 4
0%
None of these
Explanation
y = (x + 1)(x - 3) ..(1)
Substitute
y = 0
to get the point of intersection of this curve with
x-axis
0 = (x + 1) (x - 3)
\Rightarrow x = -1, 3
So the points are
(-1, 0)
and
(3, 0)
Now differentiating eq. (1), we get
\displaystyle \dfrac{dy}{dx}=\left ( x-3 \right )+\left ( x+1 \right )=2x-2
\Rightarrow \displaystyle \left ( \dfrac{dy}{dx} \right )=2\left ( x-1 \right )
Thus slope at
(-1,0)
is
=\left (\dfrac{dy}{dx} \right )_{\left ( -1,0 \right )}=-4
and at
(3,0)
is
=\displaystyle \left ( \dfrac{dy}{dx} \right )_{\left ( 3,0 \right )}=4
Hence, option 'C' is correct.
The equation of tangent at the point
\displaystyle \left ( at^{2},at^{3} \right )
on the curve
\displaystyle ay^{2}=x^{3}
is
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\displaystyle 3tx-2y=at^{3}
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\displaystyle tx-3y=at^{3}
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\displaystyle 3tx+2y=at^{3}
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None of these
The equation of the normal to the curve
\left (\displaystyle x=at^{2} \right ), \left (y=2at \right )
at '
t
' point is -
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\displaystyle ty=x+at^{2}
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\displaystyle y+tx-2at-at^{3}=0
0%
\displaystyle y=tx-2at-at^{3}
0%
None of these
Explanation
Given
\displaystyle x=at^{2},y=2at
\Rightarrow \cfrac{dx}{dt}=2at, \cfrac{dy}{dt}=2a
Thus slope of normal at point 't' is
=-\cfrac{dx}{dy}=\dfrac{dx/dt}{dy/dt}=-t
Therefore, the equation of normal at
't'
is given by,
(y-2at)=-t(x-at^2)
\Rightarrow y+tx-2at-at^3=0
Hence, option 'B' is correct.
The equation of the tangent to the curve is
\displaystyle y=2\sin x+\sin 2x
at the point
\displaystyle x=\dfrac {\pi }3
is -
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\displaystyle 2y=\sqrt{3}
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\displaystyle 3y=\sqrt{2}
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\displaystyle 2y=3\sqrt{3}
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2y = 3
Explanation
\displaystyle y=2\sin x+\sin 2x
\displaystyle \frac{dy}{dx}=2\cos x+2\cos 2x
The slope of tangent at
x=\cfrac{\pi}{3}
is,
m=\displaystyle \left ( \dfrac{dy}{dx} \right )_{x=\dfrac{\pi }{3}}=2\times \dfrac{1}{2}+2\times \left ( -\dfrac{1}{2} \right )=1-1=0
Now at
\displaystyle x=\dfrac{\pi }{3};\: \: \: y=2\times \dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{3}}{2}
\displaystyle y=\dfrac{\sqrt{3}}{2}\times 3
So the point is
\displaystyle \left ( \frac{\pi }{3},\frac{3\sqrt{3}}{2} \right )
Therefore, required tangent is,
\displaystyle \left ( y-\dfrac{3\sqrt{3}}{2} \right )=0
\Rightarrow y=\dfrac{3\sqrt{3}}{2}
\Rightarrow 2y=3\sqrt{3}
Hence, option 'C' is correct.
The equation of the tangent to the curve
\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}
at the point
\displaystyle \left ( x_{1},y_{1} \right )
is -
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\displaystyle \frac{x}{\sqrt{x_{1}}}+\frac{y}{\sqrt{y_{1}}}=\frac{1}{\sqrt{a}}
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\displaystyle \frac{x}{\sqrt{x_{1}}}+\frac{y}{\sqrt{y_{1}}}=\sqrt{a}
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\displaystyle x\sqrt{x_{1}}+y\sqrt{y_{1}}=\sqrt{a}
0%
None of these
Explanation
\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}
at point
\displaystyle \left ( x_{1},y_{1} \right )
\displaystyle \frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{y}}\frac{dy}{dx}=0
\displaystyle \frac{dy}{dx}=-\sqrt{\frac{y}{x}}\: \: \Rightarrow \: \: \left ( \frac{dy}{dx} \right )_{\left ( x_{1},y_{1} \right )}=-\frac{\sqrt{y_{1}}}{\sqrt{x_{1}}}
Thus equation of tangent is given by,
\displaystyle y-y_{1}=-\frac{\sqrt{y_{1}}}{\sqrt{x_{1}}}\left ( x-x_{1} \right )
\displaystyle y\sqrt{x_{1}}-y_{1}\sqrt{x_{1}}=-x\sqrt{y_{1}}+x_{1}\sqrt{y_{1}}
\displaystyle x\sqrt{y_{1}}+y\sqrt{x_{1}}=\sqrt{x_{1}}\sqrt{y_{1}}\left ( \sqrt{a} \right )
\displaystyle \frac{x}{\sqrt{x_{1}}}+\frac{y}{\sqrt{y_{1}}}=\sqrt{a}
Hence, option 'B' is correct.
The coordinates of the point on the curve
\displaystyle y=x^{2}+3x+4
the tangent at which passes through the origin are -
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(-2, 2), (2, 14)
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(1, -1), (3, 4)
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(2, 14), (2, 2)
0%
(1, 2), (14, 3)
Explanation
Let the point is
P(a,b)
Now the given curve is
y=x^2+3x+4
Differentiating w.r.t
x
\cfrac{dy}{dx}=2x+3
Thus slope of tangent at P is
=\left(\cfrac{dy}{dx}\right)_{(a,b)}=2a+3
Hence equation of tangent at P is given by,
(y-b)=(2a+3)(x-a)
But given that tangent passes through origin
\Rightarrow (0-b)=(2a+3)(0-a)\Rightarrow 2a^2+3a=b ..(1)
Also the point P lies on the given curve,
b=a^2+3a+4 ...(2)
Solving (1) and (2) we get the required point P coordinate
which are
(-2,2) \mbox{or} (2,14)
Hence, option 'A' is correct.
At what point the tangent line to the curve
\displaystyle y=\cos \left ( x+y \right ),\left ( -2\pi \leq x\leq 2\pi \right )
is parallel to
x + 2y = 0
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\displaystyle \left ( \dfrac {\pi }2, 0 \right )
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\displaystyle \left ( -\dfrac {\pi }2, 0 \right )
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\displaystyle \left (\dfrac{ 3\pi }2, 0 \right )
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\displaystyle \left (-\dfrac{ 3\pi }2,\dfrac { \pi }2 \right )
Explanation
\displaystyle y=\cos \left ( x+y \right )
\displaystyle \left ( -2\pi \leq x\leq 2\pi \right )
\Rightarrow \displaystyle \frac{dy}{dx}=-\sin \left ( x+y \right )\left ( 1+\frac{dy}{dx} \right )
\Rightarrow \displaystyle \frac{dy}{dx}\left ( 1+\sin \left ( x+y \right ) \right )=-\sin \left ( x+y \right )
\Rightarrow \displaystyle \frac{dy}{dx}=-\frac{\sin \left ( x+y \right )}{1+\sin \left ( x+y \right )}
Given tangent is parallel to the line,
x + 2y = 0
\Rightarrow \displaystyle \frac{dy}{dx}=-\frac{\sin \left ( x+y \right )}{1+\sin \left ( x+y \right )}=-\frac{1}{2}
\displaystyle 2\sin \left ( x+y \right )=1+\sin \left ( x+y \right )
\displaystyle \sin \left ( x+y \right )=1\Rightarrow \cos(x+y)=0\Rightarrow x+y=\cfrac{\pi}{2}
Also the point lies on the given curve,
y=0\Rightarrow x= \dfrac{\pi }{2}
Thus the point is,
\displaystyle \left ( \frac{\pi }{2},0 \right )
Hence, option 'A' is correct.
The point at which the tangent to the curve
\displaystyle y=x^{3}+5
is perpendicular to the line
x + 3y = 2
are
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(6, 1), (-1, 4)
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(6, 1) (4, -1)
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(1, 6), (1, 4)
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(1, 6), (-1, 4)
Explanation
Let point be
\displaystyle \left ( x_{1},y_{1} \right )
y = \displaystyle x^{3}+5
\Rightarrow \displaystyle \dfrac{dy}{dx}=3x^{2}
Now, the slope of tangent at this point is
=\displaystyle \left ( \frac{dy}{dx} \right )_{x_{1}y_{1}}=3x_{1}^{2}
It is
\displaystyle \perp
to
x + 3y - 2 = 0
So
\displaystyle 3x_{1}^{2}\times -\frac{1}{3}=-1\Rightarrow x_1=\pm 1
Thus,
at
\displaystyle x_{1}=1
,
\Rightarrow
\displaystyle y_{1}=6
and at
\displaystyle x_{1}=-1
\Rightarrow
\displaystyle y_{1}=4
Thus, the points are
(1, 6)
and
(-1, 4)
Hence, option 'D' is correct.
A normal
P(x,y)
on a curve meets the
X-
axis at
Q
and
N
is the ordinate at
P
.
If
NQ=\dfrac {x(1+y^2)}{1+x^2}
.
Then the equation of curves passing through
(3,1)
is
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5(1+y^2)=(1+x^2)
0%
5(1+y^2)=5(1+x^2)
0%
5(1+x^2)=(1+y^2)
0%
None of these
At what point of the curve
\displaystyle y=2x^{2}-x+1
tangent is parallel to
y = 3x + 4
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(0, 1)
0%
(1, 2)
0%
(-1, 4)
0%
(2, 7)
Explanation
Let the point is
P(a,b)
Now the given curve is
y=2x^2-x+1
Differentiating w.r.t
x
\cfrac{dy}{dx}=4x-1
Thus slope of tangent at P is
=\left(\cfrac{dy}{dx}\right)_{(a,b)}=4a-1
But given the tangent is parallel to line
y=3x+4
\Rightarrow 4a-1=3\Rightarrow a=1
Also the point P lies on the given curve,
b=2a^2-a+1=2
Therefore, the point P is
(1,2)
Hence, option 'B' is correct.
Tangents are drawn from origin to the curve
\displaystyle y=\sin x
then point of contect lies on -
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\displaystyle x^{2}=y^{2}
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\displaystyle x^{2}y^{2}=0
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\displaystyle x^{2}y^{2}=x^{2}-y^{2}
0%
None of these
Explanation
y=\sin x
\Rightarrow \cfrac{dy}{dx}=\cos x
Let the point of contact of tangent passing through origin to the given curve is
P(h,k)
So equation of tangent to this curve passes through
(0,0)
is,
y=(\cos h) x
For point of contact this line will pass through point P,
\Rightarrow k=h\cos h\Rightarrow \cos h=\cfrac{k}{h} ...(1)
Also point P lies on the given curve,
\sin h=k ..(1)
Now using
(1)^2+(2)^2=1
\cfrac{k^2}{h^2}+k^2=1\Rightarrow k^2h^2=h^2-k^2
Thus locus of
P(h,k)
is,
x^2y^2=x^2-y^2
Hence, option 'C' is correct.
A tangent to the curve
\displaystyle y=x^{2}+3x
passes through a point
(0, -9)
if it is drawn at the point -
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(-3, 0)
0%
(1, 4)
0%
(0, 0)
0%
(-4, 4)
Explanation
Let the point is
\displaystyle P\left ( x_{1},y_{1} \right )
on the curve
So,
\displaystyle \frac{dy}{dx}=2x+3
\Rightarrow \displaystyle \left ( \frac{dy}{dx} \right )_{\left ( x_{1},y_{1} \right )}=2x_{1}+3
Therefore, the tangent is
\displaystyle \frac{y-y_{1}}{x-x_{1}}=2x_{1}+3
\displaystyle \left ( y-y_{1} \right )=\left ( x-x_{1} \right )\left ( 2x_{1}+3 \right )
\displaystyle y-y_{1}=2xx_{1}+3x-2x_{1}^{2}-3x_{1}
\displaystyle x\left ( 2x_{1}+3 \right )-y+y_{1}-2x_{1}^{2}-3x_{1}=0
It passes
(0, -9)
then
\displaystyle 0\times \left ( 2x_{1}+3 \right )+9+y_{1}-2x_{1}^{2}-3x_{1}=0
\displaystyle \left ( x_{1},y_{1} \right )
also lies on the curve
=\displaystyle y_{1}-2x_{1}^{2}-3x_{1}+9=0
..........(1)
So
\displaystyle y_{1}=x_{1}^{2}+3x_{1}
..........(2)
From eq (1) and (2), we get
\displaystyle y_{1}-2x_{1}^{2}-3x_{1}+9-y_{1}+x_{1}^{2}+3x_{1}=0
\displaystyle -x_{1}^{2}+9=0
\displaystyle x_{1}^{2}=9
\displaystyle x_{1}=\pm 3
when
\displaystyle x_{1}= 3
\Rightarrow
\displaystyle y_{1}= 9+9=18
\displaystyle x_{1}= -3
\Rightarrow
\displaystyle y_{1}= 9-9=0
Thus the required points are
(-3, 0)
and
(3, 18)
Hence, option 'A' is correct.
The equation of the normal to the curve
\displaystyle y^{2}=x^{3}
at the point whose abscissa is
8
is -
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\displaystyle x\pm \sqrt{2}y=104
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\displaystyle x\pm 3\sqrt{2}y=104
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\displaystyle 3\sqrt{2}x\pm y=104
0%
None of these
Explanation
Solving the curve at
x=8=2^3
\displaystyle y^{2}=x^{3}=2^{9}
\Rightarrow \displaystyle y=\pm 2^{4}\sqrt{2}=\pm 16\sqrt{2}
So the points are
\displaystyle \left ( 8,\pm 16\sqrt{2} \right )
Now the curve is,
\displaystyle y^{2}=x^{3}
\Rightarrow \displaystyle 2y\times \frac{dy}{dx}=3x^{2}
\Rightarrow \displaystyle \left ( \frac{dy}{dx} \right )=\frac{3x^{2}}{2y}
\Rightarrow \displaystyle \left ( \frac{dy}{dx} \right )_{\left ( 8,\pm 16\sqrt{2} \right )}=\frac{3\times 64}{\pm 2\times 16\sqrt{2}}=\pm 3\sqrt{2}
Thus the slope of normal is
=\displaystyle \left ( \frac{dy}{dx} \right )=\mp \frac{1}{3\sqrt{2}}
Therefore, equation of normal is,
\displaystyle \left ( y\mp 16\sqrt{2} \right )=\mp \frac{1}{3\sqrt{2}}\left ( x-8 \right )
\Rightarrow \displaystyle \mp 3\sqrt{2}y\pm 96=(x-8)
\Rightarrow \displaystyle x\pm 3\sqrt{2}y=104
Hence, option 'B' is correct.
The normal to the curve
\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}
is perpendicular to
x
axis at the point
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(0, a)
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(a, 0)
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(\dfrac a 4, \dfrac a 4)
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No where
Explanation
Let the point be
P\displaystyle \left ( x_{1},y_{1} \right )
Now
\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{a}
Differentiating w.r.t
x
\displaystyle \frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{y}}\times \frac{dy}{dx}=0
\displaystyle \frac{dy}{dx}=-\sqrt{\frac{y}{x}}
Thus, slope of normal at P is
=\displaystyle -( \frac{dx}{dy} )_{( x_{1},y_{1} )}=-\sqrt{\frac{x_{1}}{y_{1}}}
If normal
\displaystyle \perp
to x axis then
\displaystyle \frac{dx}{dy}=\frac{a}{0}
\displaystyle \therefore y_1 = a\Rightarrow x_1 = 0
Therefore, required point is
(a, 0)
Hence, option 'B' is correct.
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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers
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