CBSE Questions for Class 11 Commerce Applied Mathematics Tangents And Its Equations Quiz 9 - MCQExams.com

Tangents are drawn from a point on the circle $$x^2+y^2=25$$ to the ellipse $$9x^2+16y^2-144=0$$ then the angle between the tangents is 
  • $$\frac{\pi}{4}$$
  • $$\frac{3\pi}{4}$$
  • $$\frac{\pi}{2}$$
  • $$\frac{2\pi}{3}$$
Slope of the line $$ \sqrt { { x }^{ 2 }+{ 4y }^{ 2 }-4xy+4 } +x-2y=1$$ equals to
  • $$ \dfrac { 1 }{ 2 }$$
  • $$2$$
  • $$ -\dfrac { 1 }{ 2 }$$
  • $$None\ of\ these$$
Let f be a real-valued differentiable function on R (the set of all real numbers) such that $$f(1)=1$$. If the y-intercept of the tangent at any point P(x, y) on the curve $$y=f(x)$$ is equal to the cube of the abscissa of P, then the value of $$f(-3)$$ is equal to?
  • $$-3$$
  • $$3$$
  • $$9$$
  • $$-9$$
The inclination of the tangent at $$\theta = \dfrac {\pi}{3}$$ on the curve $$x = a(\theta + \sin \theta), y = a(1 + \cos \theta)$$ is
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{6}$$
  • $$\dfrac {2\pi}{3}$$
  • $$\dfrac {5\pi}{6}$$
Equation of a normal to the curve $$y=x\log{x}$$, parallel to $$2x-2y+3=0$$ is
  • $$x+y=3e^{-2}$$
  • $$x-y=3e^{-2}$$
  • $$x-y=3e^{2}$$
  • $$x+y=3e^{2}$$
The greatest slope among the lines represented by the equation $$4x^2 - y^2 + 2y - 1 = 0 $$ is - 
  • $$-3$$
  • $$-2$$
  • $$2$$
  • $$3$$
The ordinate of all points on the curve $$y=\dfrac{1}{2\sin^{2}x+3\cos^{2}x}$$  where the tangent is horizontal, is
  • Always equal to $$\dfrac{1}{2}$$
  • Always equal to $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$ or $$\dfrac{1}{3}$$ according as $$n$$ is an even or an odd integer.
  • $$\dfrac{1}{2}$$ or $$\dfrac{1}{3}$$ according as $$n$$ is an even integer
Equation of a tangent to the curve $$y=\cos(x+y),\ 0\le x\le 2\pi$$ that is parallel to the line $$x+2y=0$$ is
  • $$x+2y=\pi/2$$
  • $$x+2y=\pi/4$$
  • $$x+2y=\pi$$
  • $$x+y=\pi$$
The curve given by $$x + y = {e^{xy}}$$ has an tangents parallel to the y-axis at the point
  • $$(0,1)$$
  • $$(1,0)$$
  • $$(1,1)$$
  • $$(0,0)$$
The equation of normal to the curve $$x^{3}+y^{3}=8xy$$ at points where it is meet by the curve $$y^{2}=4x$$,other then origin is
  • $$y=x$$
  • $$y=-x+4$$
  • $$y=2x$$
  • $$y=-2x$$
Tangents are drawn from origin to the curve $$y=\sin{x}+\cos{x}$$. Then their points of contact lie on the curve
  • $$\dfrac{1}{x^{2}}+\dfrac{2}{y^{2}}=1$$
  • $$\dfrac{1}{x^{2}}-\dfrac{2}{y^{2}}=-1$$
  • $$\dfrac{2}{x^{2}}+\dfrac{2}{y^{2}}=1$$
  • $$\dfrac{2}{x^{2}}-\dfrac{2}{y^{2}}=1$$
The slope of normal to the curve y= log (logx) at x = e is 
  • e
  • -e
  • $$\frac{1}{e}$$
  • -$$\frac{1}{e}$$

Number of possible tangents to the curve $$y = \cos \left( {x + y} \right), - 3\pi  \leqslant x \leqslant 3\pi $$, that are parallel to the line $$x + 2y = 0$$, is

  • 1
  • 2
  • 3
  • 4
The normal to the curve, $${x}^{2}+2xy-{3y}^{2}=0,\ at\left (1,1\right)$$:
  • Meets the curve, again in the fourth quadrant
  • Does not meet the curve again
  • Meets the curve again in the second quadrant
  • Meets the curve again in the third quadrant
Number of tangents drawn from the point $$\left (-1/2,0\right)$$ to the curve $$y={e}^{x}$$. (Here { } denotes fractional part function ). 
  • $$2$$
  • $$1$$
  • $$3$$
  • $$4$$
If the tangent at $$(x_{1}, y_{1})$$ to the curve $$x^{3}+y^{3}=a^{3}$$ meets the curve again at $$(x_{2}, y_{2})$$ then
  • $$\dfrac{x_{2}}{x_{1}}+\dfrac{y_{2}}{y_{1}}=-1$$
  • $$\dfrac{x_{2}}{y_{1}}+\dfrac{x_{1}}{y_{2}}=-1$$
  • $$\dfrac{x_{1}}{x_{2}}+\dfrac{y_{1}}{y_{2}}=-1$$
  • $$\dfrac{x_{2}}{x_{1}}+\dfrac{y_{2}}{y_{1}}=1$$
If the tangent at P of the curve $$y^2=x^3$$ intersects the curve again at Q and the straight lines OP, OQ ma angles $$\alpha, \beta$$ with the x-axis where 'O' is the origin then $$\tan\alpha/\tan\beta$$ has the value equal to?
  • $$-1$$
  • $$-2$$
  • $$2$$
  • $$\sqrt{2}$$
A curve C has the property that if the tangent drawn at any point 'P' on C meets the coordinate axes at A and B, and P is midpoint of AB. If the curve passes through the point $$(1, 1)$$ then the equation of the curve is?
  • $$xy=2$$
  • $$xy=3$$
  • $$xy=1$$
  • $$xy=4$$
The area of the triangle formed by the coordinate axes and a tangent to the curve $$xy={a}^{2}$$ at the point $$({x}_{1},{y}_{1})$$ is
  • $$\cfrac{{a}^{2}{x}_{1}}{{y}_{1}}$$
  • $$\cfrac{{a}^{2}{y}_{1}}{{x}_{1}}$$
  • $$2{a}^{2}$$
  • $$4{a}^{2}$$
If $$x={t}^{2}$$ and $$y=2t$$, then equation of the normal at $$t=1$$ is
  • $$x+y-3=0$$
  • $$x+y-1=0$$
  • $$x+y+1=0$$
  • $$x+y+3=0$$
The equation of the tangent to curve $$\sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=2$$ at the point (a, b) is 
  • $$\frac{x}{a}-\frac{y}{b}=0$$
  • $$\frac{x}{a}+\frac{y}{b}=2$$
  • $$\frac{x}{a}-\frac{y}{b}=1$$
  • $$\frac{x}{a}+\frac{y}{b}=0$$
The equation of tangent to the ellipse $$4{x^2} + 9{y^2} = 36$$ at $$(3, - 2)\,is\,$$
  • $$2x - 3y = 6$$
  • $$3x - 2y = 13$$
  • $$x + y = 1$$
  • $$x - y = 5$$
The equation of the normal to the curve $${x}^{4}=4y$$ through the point $$(2,4)$$ is
  • $$x+8y=34$$
  • $$x-8y+30=0$$
  • $$8x-2y=0$$
  • $$8x+y=20$$
The equation of tangents to the ellipse $${x^2} + 4{y^2} = 25$$ at the point whose ordinate is 2, is 
  • $$3x + 8y - 25 = 0$$
  • $$-3x + 8y = - 25$$
  • $$-3x - 8y = 25$$
  • $$3x + 8y = 35$$
The equation of tangent to the curve $$y=x^{2}+4x+1$$ at $$\left(-1,-2\right)$$ is
  • $$2x+y-5=0$$
  • $$2x-y=0$$
  • $$2x-y-1=0$$
  • $$x+y-1=0$$
Two lines drawn through the point $$A ( 4,0 )$$  divide the area bounded by the curve $$y = \sqrt { 2 } \sin ( \pi x / 4 )$$  and  $$x$$ - axis between the lines $$x = 2$$  and   $$x = 4$$  into three equal parts. Sum of the slopes of the drawn lines is:
  • $$- 4 \sqrt { 2 } / \pi$$
  • $$- \sqrt { 2 } / \pi$$
  • $$- 2 \sqrt { 2 } / \pi$$
  • None
The tangent to the curve $$2a^2y=x^3-3ax^2$$ is parallel to the x-axis at the points
  • $$(0 , 0) , (2a, - 2a)$$
  • $$(0 , 0) , (-2a,  2a)$$
  • $$(0 , 0) , (-2a, - 2a)$$
  • $$(2 , 2) , (0 , 0)$$
If $$x-2y+k=0$$ is a common tangent to $$\displaystyle{ y }^{ 2 }=4x\quad \& \frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { 3 } } =1\left( a>\sqrt { 3 }  \right)  $$, then the value of a, k and other common tangent are given by
  • $$a = 2$$
  • $$a = -2$$
  • $$x+2y+4 = 0$$
  • $$k=4$$
The point on the curve $${x}^{2}+{y}^{2}-2x-3=0$$ at which the tangent in parallel to x-axis is 
  • $$(1,0),(-1,-4)$$
  • $$(0,-1),(-2,3)$$
  • $$(2,13),(-2,-3)$$
  • $$(1,2),( 1,-2)$$
Slope of tangent to the circle $$( x - r ) ^ { 2 } + y ^ { 2 } = r ^ { 2 }$$ at the point $$( x , y )$$ lying on the circle is
  • $$\frac { x } { y - r }$$
  • $$\frac { r - x } { y }$$
  • $$\frac { y ^ { 2 } - x ^ { 2 } } { 2 x y }$$
  • $$\frac { y ^ { 2 } + x ^ { 2 } } { 2 x y }$$
If the slope of one of the lines represented $${a^3}{x^2} + 2hxy + {b^3}{y^2} = 0$$ be the square of the other, then $$ab(a+b)$$ is equal to:
  • $$2h$$
  • $$-2h$$
  • $$8h$$
  • $$-8h$$
The normal to the curve, $${{\text{x}}^{\text{2}}}{\text{ + 2xy - 3}}{{\text{y}}^{\text{2}}}{\text{ = 0,}}\;{\text{at}}\;\left( {{\text{1,1}}} \right){\text{:}}$$
  • meets the curve again in the second quadrant.
  • meets the curve again in the third quadrant.
  • meets the curve again in the fourth quadrant.
  • does not meet the curve again.
The line $$3x-4y=0$$
  • is a tangent to the circle $${x}^{2}+{y}^{2}=25$$
  • is a normal to the circle $${x}^{2}+{y}^{2}=25$$
  • does not meet the circle $${x}^{2}+{y}^{2}=25$$
  • does not pass thro' the origin
 The curves $$x = y^2   and.  xy = k$$ cut at right angles, If $$6k^2$$ = 1.
  • True
  • False
The equation of the normal to the curve $$y^4=ax^3$$ at (a , a) is 
  • x + 2y=3a
  • 3x-4y+a=0
  • 4x+3y=7a
  • 4x-3y=a
Which of the following lines, is a normal to the parabola $${y}^{2}=16x$$?
  • $$y=x-11\cos{\theta}-3\cos{3\theta}$$
  • $$y=x-11\cos{\theta}-\cos{3\theta}$$
  • $$y=(x-11)\cos{\theta}+\cos{3\theta}$$
  • $$y=(x-11)\cos{\theta}-\cos{3\theta}$$
The equation of one of the tangents to the curve $$y=\cos(x+y),-2\pi 
\le x \le 2\pi$$; that is parallel to the line $$x+ 2y = 0$$ , is
  • $$x+2y=1$$
  • $$x+2y=\dfrac {\pi}{2}$$
  • $$x+2y=\dfrac {\pi}{4}$$
  • $$None\ of\ these$$
The tangent at any point of the curve $$x={ at }^{ 3 },y={ at }^{ 4 }$$ divides the abscissa of the point of contact in the ratio
  • $$1:4$$
  • $$3:2$$
  • $$1:3$$
  • $$3:1$$
State true or false.
The curves $$y = {x^2} - 3x + 1$$ and $$x\left( {y + 3} \right) = 4$$ intersect at the right angles at their point of intersection
  • True
  • False
The slope of the straight line which is both tangent and normal to the curve $$4x^3=27y^2$$ is 
  • $$\underline { + } 1$$
  • $$\underline { + } \dfrac { 1 }{ 2 } $$
  • $$\underline { + } \dfrac { 1 }{ \sqrt { 2 } } $$
  • $$\underline { + } \sqrt { 2 } $$
The angle between the curves $$y = \sin x$$ and $$y = \cos x$$ is 
  • $$\tan^{-1}(2\sqrt{2})$$
  • $$\tan^{-1}(3\sqrt{2})$$
  • $$\tan^{-1}(3\sqrt{3})$$
  • $$\tan^{-1}(5\sqrt{2})$$
The normal to the curve $$x=\quad a(cos\theta +\theta sin\theta ),\quad y=\quad a(sin\theta -\theta cos\theta )$$ at any point $$'\theta '$$ is such that
  • it passes through the origin
  • it makes an angle $$\dfrac { \pi }{ 2 } +\theta $$ with the x-axis
  • it is at a constant distance from the origin 
  • it passes through $$\left(a,\dfrac{\pi}{2}\right)$$
The tangent to the curve $$y=e^{2x}$$ at the point $$(0, 1)$$ meets x-axis at?
  • $$(0, 1)$$
  • $$\left(-\dfrac{1}{2}, 0\right)$$
  • $$(2, 0)$$
  • $$(0, 2)$$
Three normals are drawn from the point $$\left(c,0\right)$$ to the curve $${y}^{2}=x.$$If two of the normals are perpendicular to each other,then $$c=$$
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{4}$$
  • $$1$$
The equation to the normal to the curve $$y=\sin x$$ at $$(0, 0)$$ is
  • $$x=0$$
  • $$y=0$$
  • $$x+y=0$$
  • $$x-y=0$$
The tangent to the curve, $$y = xe^{x^2}$$ passing through the point $$(1, e)$$ also passes through the point:
  • $$\left(\dfrac{4}{3}, 2e\right)$$
  • $$(2, 3e)$$
  • $$\left(\dfrac{5}{3}, 2e\right)$$
  • $$(3, 6e)$$
If the line $$x+y=0$$ touches the curve $$2y^2=\alpha x^2+\beta $$ at $$(1,-1),$$ then $$(\alpha ,\beta )=$$
  • $$(-2,4)$$
  • $$(-1,3)$$
  • $$(4,-2)$$
  • $$(2,0)$$
Equation of the tangent at (1, -1) to the curve
$${ x }^{ 3 }-x{ y }^{ 2 }-4{ x }^{ 2 }-xy+5x+3y+1=0$$ is 
  • $$x-4y-5=0$$
  • $$x+1=0$$
  • $$y-1=0$$
  • $$y+1=0$$
The angle made by the tangent at any point on the curve $$x=a(t+\sin { t } \cos { t } ),y=a{ (1+\sin { t } ) }^{ 2 }$$ with x-axis is
  • $$\dfrac { \pi }{ 2 } $$
  • $$\dfrac { \pi }{ 4 } $$
  • $$\pi +\dfrac { t }{ 2 } $$
  • $$\dfrac { \pi }{ 4 } +\dfrac { t }{ 2 } $$
Length of the normal to the curve at any point on the curve $$y=\dfrac { a\left( { e }^{ x/a }+{ e }^{ -x/a } \right)  }{ 2 } $$ varies as 
  • $$x$$
  • $${ x }^{ 2 }$$
  • $$y$$
  • $${ y }^{ 2 }$$
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