CBSE Questions for Class 11 Engineering Maths Conic Sections Quiz 5 - MCQExams.com

The equation of the circle whose two diameters are the lines $$x+y=4$$ and $$x-y=2$$ and which passes through $$(4 , 6 )$$ is 
  • $$x^2+y^2-6x-2y-16=0$$
  • $$x^2+y^2-6x-2y=15$$
  • $$x^2+y^2=0$$
  • $$5(x^2+y^2)-4x=16$$
If the eccentricities of the hyperbola $$\dfrac {x^{2}}{a^{2}} - \dfrac {y^{2}}{b^{2}} = 1$$ and $$\dfrac {y^{2}}{b^{2}} - \dfrac {x^{2}}{a^{2}} = 1$$ be $$e$$ and $$e_{1}$$, then $$\dfrac {1}{e^{2}} + \dfrac {1}{e_{1}^{2}} =$$.
  • $$1$$
  • $$2$$
  • $$3$$
  • None of these
The equation of the circle having $$x-y-2=0$$ and $$x-y+2=0$$ as two tangents and $$x-y=0$$ as diameter is
  • $${ x }^{ 2 }+{ y }^{ 2 }+2x-2y+1=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }-2x+2y-1=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }=2$$
  • $${ x }^{ 2 }+{ y }^{ 2 }=1$$
The equation of the latus rectum of the parabola $$x^{2} + 4x + 2y = 0$$ is
  • $$2y + 3 = 0$$
  • $$3y = 2$$
  • $$2y = 3$$
  • $$3y + 2 = 0$$
The curve represented by $$x = 3\cos t + 3\sin t$$ and $$y = 4\cos t - 4\sin t$$ is
  • An ellipse
  • A hyperbola
  • A parabola
  • None of these
Find the equation of the circle which touches the coordinate axes and whose centre lies on the line $$x - 2y = 3$$.
  • $$(x-1)^2+(y+1)^2=1$$
  • $$(x+1)^2+(y+1)^2=1$$
  • $$(x-1)^2+(y-1)^2=1$$
  • $$(x+1)^2+(y-1)^2=1$$
Consider a rigid square $$ABCD$$ as in the figure with $$A$$ and $$B$$ on the $$x$$ and $$y$$ axis respectively. When $$A$$ and $$B$$ slide along their respective axes, the locus of $$C$$ forms a part of.
739522_533fd921afe8461f93ef01b1a15ae1ac.png
  • A circle
  • A parabola
  • A hyperbola
  • An ellipse which is not a circle
Suppose the parabola $$(y - k)^{2} = 4(x - h)$$ with vertex $$A$$, passes through $$O = (0, 0)$$ and $$L = (0, 2)$$. Let $$D$$ be an end point of the latus rectum. Let the y-axis intersect the axis of the parabola at $$P$$. Then $$\angle PDA$$ is equal to
  • $$\tan^{-1} \dfrac {1}{19}$$
  • $$\tan^{-1} \dfrac {2}{19}$$
  • $$\tan^{-1} \dfrac {4}{19}$$
  • $$\tan^{-1} \dfrac {8}{19}$$
If $$ e$$ and $$e'$$ be the eccentricities of a hyperbola and its conjugate, them $$ \dfrac {1}{e^2} + \dfrac {1}{e'^{2}} $$ is equal to :
  • 0
  • 1
  • 2
  • None of these
$$Ay^{2} + By + Cx + D = 0, (ABC\neq 0)$$ be the equation of a parabola, then
  • Then length of the latus rectum is $$\left |\dfrac {C}{A}\right |$$
  • The axis of the parabola is vertical
  • The y-coordinate of the vertex is $$-\dfrac {B}{2A}$$
  • The x-coordinate of the vertex is $$\dfrac {D}{A} + \dfrac {B^{2}}{4AC}$$
The equation of the circle passing through $$(2,0)$$ and $$(0,4)$$ and having the minimum radius is
  • $$x^{2}+y^{2}=20$$
  • $$x^{2}+y^{2}-2x-4y=0$$
  • $$x^{2}+y^{2}=4$$
  • $$x^{2}+y^{2}=16$$
If $$A(5,-4)$$ and $$B(7,6)$$ are points in a plane, then the set of all points $$P(x,y)$$ in the plane such that $$AP=PB=2:3$$ is
  • a circle
  • a hyperbola
  • an ellipse
  • a parabola
The equation of the circle which passes through the points $$(2, 3)$$ and $$(5)$$ and the centre lies on the straight line $$y - 4x + 3 = 0$$, is
  • $$x^{2} + y^{2} + 4x - 10y + 25 = 0$$
  • $$x^{2} - y^{2} - 4x - 10y + 25 = 0$$
  • $$x^{2} + y^{2} - 4x - 10y + 16 = 0$$
  • $$x^{2} + y^{2} - 14y + 8 = 0$$
The equation of the image of the circle $${ x }^{ 2 }+{ y }^{ 2 }+16x-24y+183=0$$ by the line mirror $$4x+7y+13=0$$ is:
  • $${ x }^{ 2 }+{ y }^{ 2 }+32x-4y+235=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }+32x+4y-235=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }+32x-4y-235=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }+32x+4y+235=0$$
The equation of the circle, which is the mirror image of the circle, $${ x }^{ 2 }+{ y }^{ 2 }-2x=0$$, in the line, $$y=3-x$$ is:
  • $${ x }^{ 2 }+{ y }^{ 2 }-6x-8y+24=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }-8x-6y+24=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }-4x-6y+12=0\quad $$
  • $${ x }^{ 2 }+{ y }^{ 2 }-6x-4y+12=0\quad $$
Point $$\left( 0,\lambda  \right) $$ lies in the interior of circle $$x^2+y^2=c^2$$ then
  • $$\lambda \in \left( -c,c\right) $$
  • $$\lambda \in \left(0,c \right) $$
  • $$\lambda \in \left(-c,0\right) $$
  • None.
If the lines $$3x - 4y - 7 = 0$$ and $$2s - 3y - 5 = 0$$ are two diameters of a circle of area $$49\pi$$ square units, the equation of the circle is-
  • $$x^{2} + y^{2} + 2x - 2y - 62 = 0$$
  • $$x^{2} + y^{2} - 2x + 2y - 62 = 0$$
  • $$x^{2} + y^{2} - 2x + 2y - 47 = 0$$
  • $$x^{2} + y^{2} + 2x - 2y - 47 = 0$$
Find the equation of the circle which passes through the point $$(1, 1)$$ & which touches the $$x^2 + y^2 + 4x - 6y - 3 = 0$$ at the point $$(2, 3)$$ on it.
  • $$x^2 + y^2 + x - 6y + 3 = 0$$
  • $$x^2 + y^2 + x - 6y - 3 = 0$$
  • $$x^2 + y^2 + x + 6y + 3 = 0$$
  • $$x^2 + y^2 + x - 3y + 3 = 0$$
If the locus of the point $$(4t^2 - 1, 8t-2)$$ represents a parabola then the equation of latus rectum is
  • $$x - 5 = 0$$
  • $$2x - 7 = 0$$
  • $$x + 5 = 0$$
  • $$x -3 = 0$$
Find the locus of the point of intersection of the lines $$\sqrt{3}x-y-4\sqrt{3} \lambda=0$$ and $$\sqrt{3}\lambda x+\lambda y-4\sqrt{3}=0$$ for different values of $$\lambda$$.
  • $$4x^2-y^2=48$$
  • $$x^2-4y^2=48$$
  • $$3x^2-y^2=48$$
  • $$y^2-3x^2=48$$
Find the equation of the circle which passes through the point (1, 1) & which touches the circle $$x^{2} + y^{2} + 4x - 6y - 3 = 0$$ at the point $$(2, 3)$$ on it.
  • $$x^{2} + y^{2} + x - 6y + 3 = 0$$
  • $$x^{2} + y^{2} + x - 6y - 3 = 0$$
  • $$x^{2} + y^{2} + x + 6y + 3 = 0$$
  • $$x^{2} + y^{2} + 4x - 3y + 3 = 0$$
The equation of the hyperbola whose foci are $$(6, 5), (-4, 5)$$ and eccentricity $$5/4$$ is?
  • $$\displaystyle\frac{(x-1)^2}{16}-\frac{(y-5)^2}{9}=1$$
  • $$\displaystyle\frac{x^2}{16}-\frac{y^2}{9}=1$$
  • $$\displaystyle\frac{(x-1)^2}{16}-\frac{(y-5)^2}{9}=-1$$
  • $$\displaystyle\frac{(x-1)^2}{4}-\frac{(y-5)^2}{9}=1$$
The equation of the parabola with vertex at (0, 0), axis along x-axis and passing through $$\displaystyle \left( \frac{5}{3}, \frac{10}{3} \right)$$ is
  • $$y^2 = 20 x$$
  • $$\displaystyle y^2 = \frac{20 x}{3}$$
  • $$\displaystyle y^2 = \frac{10 x}{3}$$
  • $$\displaystyle x^2 = \frac{20 y}{3}$$
If $$b$$ and $$c$$ are the lengths of the segments of any focal chord of a parabola $$y^{2} = 4ax$$, then length of the semi-latus rectum is
  • $$\dfrac{b+c}{2}$$
  • $$\dfrac{bc}{b+c}$$
  • $$\dfrac{2bc}{b+c}$$
  • $$\sqrt{bc}$$
If the curve y = | x- 3| touches the parabola $$y^2 = \lambda (x-4), \lambda >0$$, then latus rectum of the parabola, is
  • 2
  • 4
  • 8
  • 16
State whether following statements are true or false
Statement-1 : The only circle having radius $$\sqrt {10}$$ and a diameter along line $$2x + y=5$$ is $$x^2 + y^2 - 6x + 2y=0$$.
Statement-2: The line 2x + y=5 is a normal to the circle $$x^2+ y^2 - 6x + 2y =0.$$
  • Statement- 1 is false, statement-2 is true.
  • Statement-1 is true,statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
  • Statement-1 is true, statement-2 is false.
  • Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
If the curves $$f(x) = e^{x}$$ and $$g(x) = kx^{2}$$ touches each other then the value of $$k$$ is equal to
  • $$2$$
  • $$\dfrac {e^{2}}{4}$$
  • $$\dfrac {e^{2}}{2}$$
  • $$\dfrac {e}{2}$$
The equation of the circle passing through $$(4,\ 5)$$ having the centre at $$(2 ,\ 2)$$ is
  • $$x^{2} + y^{2} + 4x + 4y-5=0$$
  • $$x^{2} + y^{2} - 4x - 4y-5=0$$
  • $$x^{2} + y^{2} -4x = 13$$
  • $$x^{2} + y^{2} - 4x - 4y + 5=0$$
Let circles $${C}_{1}$$ and $${C}_{2}$$ an Argand plane be given by $$\left| z+1 \right| =3$$ and $$\left| z-2 \right| =7\ \ $$ respectively. If a variable circle $$\left| z-{ z }_{ 0 } \right| =r\quad $$ be inside circle $${C}_{2}$$ such that it touches $${C}_{1}$$ externally and $${C}_{2}$$ internally then locus of $${z}_{0}$$ describes a conic $$E$$ whose eccentricity is equal to
  • $$\cfrac { 1 }{ 10 } $$
  • $$\cfrac { 3 }{ 10 } $$
  • $$\cfrac { 5 }{ 10 } $$
  • $$\cfrac { 7 }{ 10 } $$
The foci of an ellipse are located at the points $$(2, 4)$$ and $$(2, -2)$$. The points $$(4, 2)$$ lies on the ellipse. If $$a$$ and $$b$$ represent the lengths of the semi-major and semi-minor axes respectively, then the value of $$(ab)^{2}$$ is equal to
  • $$68 + 22\sqrt {10}$$
  • $$6 + 22\sqrt {10}$$
  • $$26 + 10\sqrt {10}$$
  • $$6 + 10\sqrt {10}$$
state whether following statements are true or false
Statement 1 : $$\sqrt {(x-1)^2 +y^2} + \sqrt{(x+1)^2 + y^2} = 4$$ represent equation of ellipse
Statement 2 : The locus of point which moves such that sum of its distance from two fixed points is constant is an ellipse.
  • Statement 1 is true, statement 2 is true and statement 2 is correct explanation for statement 1
  • Statement 1 is true, statement 2 is true and statement 2 is NOT correct explanation for statement 1
  • Statement 1 is true, statement 2 is false
  • Statement 1 is false, statement 2 is true
Equation of the parabola having focus $$(3,2)$$ and Vertex $$(-1,2)$$ is 
  • $$(x+1)^2=16(y-2)$$
  • $$(x-1)^2=16(y+2)$$
  • $$(y-2)^2=16(x+1)$$
  • $$(y+2)^2=16(x-1)$$
The equation of the circle whose radius is $$5$$units and which touches the circle, $${x}^{2}+{y}^{2}-2x-4y-20=0$$ at the point $$\left(5,5\right)$$ is
  • $${x}^{2}+{y}^{2}+18x+16y+120=0$$
  • $${x}^{2}+{y}^{2}-18x-16y+120=0$$
  • $${x}^{2}+{y}^{2}-18x-16y-120=0$$
  • $${x}^{2}-{y}^{2}-18x-16y-120=0$$
The equation $$\dfrac{x^2}{2-a}+\dfrac{y^2}{a-5}+1=0$$ represents an ellipse if $$a\in$$
  • $$\left(2,\dfrac{3}{2}\right)\cup\left(\dfrac{3}{2},5\right)$$
  • $$\left(2,\dfrac{3}{2}\right)$$
  • $$\left(1,\dfrac{3}{2}\right)$$
  • $$\left(\dfrac{3}{2},5\right)$$
The equation of the latus rectum of the parabola $${ x }^{ 2 }+4x+2y=0$$ is
  • $$2y+3=0$$
  • $$3y=2$$
  • $$2y=3$$
  • $$3y+2=0$$
Find the equation of the circle which passes through the points $$(2,-2)$$ and $$(3,4)$$. And whose centre lies on the line $$x+y=2$$.
  • $${ x }^{ 2 }+{ y }^{ 2 }+7x+3y+16=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }+7x-3y-16=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }+4x+5y-16=0$$
  • $${ x }^{ 2 }+{ y }^{ 2 }+5x+8y+30=0$$
If $$16{m}^{2}-8l-1=0,$$ then equation of the circle having $$lx+my+1=0$$ is a tangent is
  • $${x}^{2}+{y}^{2}+8x=0$$
  • $${x}^{2}+{y}^{2}-8x=0$$
  • $${x}^{2}+{y}^{2}+8y=0$$
  • $${x}^{2}+{y}^{2}-8y=0$$
If the line $$y = x \sqrt{3} - 3$$ cuts the parabola $$y^2 = x + 2$$ at $$P$$ and $$Q$$ and if $$A$$ be the points $$(\sqrt{3}, 0)$$ , then $$AP.AQ$$ is
  • $$\dfrac{2}{3} (\sqrt{3} + 2)$$
  • $$\dfrac{4}{3} (\sqrt{3} + 2)$$
  • $$\dfrac{4}{3} (2 -\sqrt{3})$$
  • $$\dfrac{4}{6 - 3\sqrt{3}}$$
The equation of the circle having normal at $$(3, 3)$$ as $$y = x$$ and passing through $$(2, 2)$$ is:
  • $$x^2 + y^2 - 3x - 7y + 12 = 0$$
  • $$x^2 + y^2 -4x - 6y + 12 = 0$$
  • $$x^2 + y^2 - 6x - 4y + 12 = 0$$
  • $$x^2 + y^2 - 5x - 5y + 12 = 0$$

In the $$xy$$ plane,  the segment with end points$$(3,8)$$ and $$(
-5,2)$$ is the diameter of the circle. The  point $$(k,10)$$ lies on the circle for:
  • no value of $$k$$
  • exactly one integral $$k$$
  • exactly one non integral $$k$$
  • two real values of $$k$$
If the equation $$\dfrac{\lambda (x+1)^2}{3}+\dfrac{(y+2)^2}{4}=1$$ represents a circle then $$\lambda = ?$$
  • $$1$$
  • $$\dfrac{3}{4}$$
  • $$0$$
  • $$-\dfrac{3}{4}$$
A circle is dawn its centre on the line $$ x+y=2$$ to touch the line $$ 4x-3y+4=0$$ and pass through the point $$(0,1)$$. Find the equation.
  • $$(x-1)^2+(y-1)^2=1$$
  • $$(x-20)^2+(y+18)^2=841$$
  • $$(x-21)^2+(y+19)^2=841$$
  • None
If the equation $$\dfrac { \lambda { \left( x+1 \right)  }^{ 2 } }{ 3 } +\dfrac { { \left( y+2 \right)  }^{ 2 } }{ 4 }=1$$ represents a circle then $$\lambda=$$
  • $$1$$
  • $$\dfrac {3}{4}$$
  • $$0$$
  • $$-\dfrac {3}{4}$$
A circle of radius $$2$$ lies in the first quadrant and touches both the axes of co-ordinates. Then the equation of the circle with centre $$(6, 5)$$ and touching the above circle externally is
  • $$(x - 6)^2 + (y - 5)^2 = 4$$
  • $$(x - 6)^2 + (y - 5)^2 = 9$$
  • $$(x - 6)^2 + (y - 5)^2 = 36$$
  • none of these
The foci of the ellipse $$\dfrac{x^{2}}{16} + \dfrac{y^{2}}{b^{2}} =1$$ and the hyperbola $$\dfrac{x^{2}}{144} - \dfrac{y^{2}}{81} =\dfrac{1}{25}$$ coincide, then the value of $$b^{2}$$ is:
  • $$5$$
  • $$7$$
  • $$9$$
  • $$4$$
The radius of the circle $$x^2+y^2-5x+2y+5=0$$ is
  • $$2$$
  • $$1$$
  • $$\dfrac{3}{2}$$
  • $$\dfrac{2}{3}$$
The equation $$2x^2+3y^2-8x-18y+35=\lambda$$ represents?
  • A circle for all $$\lambda$$
  • An ellipse if $$\lambda < 0$$
  • The empty set if $$\lambda > 0$$
  • A-point if $$\lambda = 0$$
Coordinates of centre and radius of the circle $$(x-3)^2+(y+4)^2=25$$ are respectively
  • $$(3,4)$$ , $$25$$
  • $$(-3,4)$$ , $$5$$
  • $$(3,-4)$$ , $$5$$
  • $$(3,-4)$$ , $$25$$
The parabola $$y = px^{2} + px + q$$ is symmetrical about the line
  • $$x=q$$
  • $$x=p$$
  • $$2x=1$$
  • $$2x + 1=0$$
The magnitude of the gradient of the tangent at an extremity of latera recta of the hyperbola $$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$$ is equal to (where $$e$$ is the eccentricity of the hyperbola)
  • $$be$$
  • $$e$$
  • $$ab$$
  • $$ae$$
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