Mathematics · Quantitative Aptitude

Conic Sections

239 Questions

Conic sections deal with the curves obtained by the intersection of a cone with a plane, primarily focusing on parabolas, ellipses, and hyperbolas. Questions require finding vertices, directrices, and asymptotes based on given equations. This is an advanced geometry topic for rigorous competitive exams.

Parabola equationsHyperbola propertiesEllipse conceptsTangents and normalsDirectrix and focus

Conic Sections Questions

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

A parabola $y = ax^2 + bx + c$ crosses the x-axis at $(\alpha, 0)$ $(\beta, 0)$ both to the right of the origin. A circle also passes through these two points. The length of the tangent from the origin to the circle is

  1. $\displaystyle \sqrt{\frac{bc}{a}}$
  2. $ac^2$
  3. $\displaystyle \frac{b}{a}$
  4. $\displaystyle \sqrt{\frac{c}{a}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$OT$ is a tangent and $OAB$ is a secant 


we know that

$OT^2 =OA.OB$

         $=\alpha\beta$

         $=\dfrac{c}{a}$ (Since $\alpha,\beta $ are the roots of $y=ax^2+bx+c$)

$\Rightarrow OT=\sqrt{\dfrac{c}{a}}$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Let ${P} _{1}$ and ${P} _{2}$ be two fixed points in $xy-plane$. A line ${L} _{1}=0$ passes through ${P} _{1}$ intersects $y-axis$ at $B$ and the line ${L} _{2}=0$ passes through ${P} _{2}$ and intersects $x-axis$ at $A$. If ${L} _{1}=0$ and ${L} _{2}=0$ are perpendicular then the locus of mid-point of$AB$ is

  1. $Straight line$
  2. $Circle$
  3. $Ellipse$
  4. $Parabola$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The locus of the midpoint of a segment whose endpoints move on axes while the lines are perpendicular results in a straight line relationship based on the fixed points P1 and P2.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The locus of the mid-point of that chord of parabola which subtends right angle on the vertex will be

  1. $y ^ { 2 } - 2 a x + 8 a ^ { 2 } = 0$
  2. $y ^ { 2 } = a ( x - 4 a )$
  3. $y ^ { 2 } = 4 a ( x - 4 a )$
  4. $y ^ { 2 } + 3 a x + 4 a ^ { 2 } = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a parabola y^2 = 4ax, the chord subtending a right angle at the vertex has the equation y = mx + 2am. The midpoint (h, k) of this chord satisfies k = mh + 2am and the property that the chord is y = (2a/k)x - 4a^2/k. Substituting and simplifying leads to the locus y^2 = 2a(x - 4a), which is y^2 - 2ax + 8a^2 = 0.

Multiple choice maths ellipse normal to an ellipse tangent and normal to an ellipse two dimensional analytical geometry-ii

If the tangent drawn at a point $\left( { t }^{ 2 },2t \right) $ on the parabola ${ y }^{ 2 }=4x$ is same as normal drawn at $\left( \sqrt { 5 } \cos { \alpha  } ,2\sin { \alpha  }  \right) $ on the ellipse $\displaystyle \frac { { x }^{ 2 } }{ 5 } +\frac { { y }^{ 2 } }{ 4 } =1$, then which of following is true.

  1. $\displaystyle t=\pm \frac { 1 }{ \sqrt { 5 } } $
  2. $\alpha =-\tan ^{ -1 }{ 2 } $
  3. $\alpha =\tan ^{ -1 }{ 2 } $

  4. None of these
Reveal answer Fill a bubble to check yourself
A,B,C Correct answer
Explanation

Equation of tangent to ${ y }^{ 2 }=4x$ at $\left( { t }^{ 2 },2t \right) $ is $x=ty-{ t }^{ 2 }$   ....(1)


Equation of normal to ellipse $\displaystyle \frac { { x }^{ 2 } }{ 5 } +\frac { { y }^{ 2 } }{ 4 } =1$ at $\left( \sqrt { 5 } \cos { \alpha  } ,2\sin { \alpha  }  \right) $ is $\sqrt { 5 } \sec { \alpha x-2y\csc { \alpha  } =1 } $    ....(2)

Given (1) $=$ (2)

$\displaystyle \Rightarrow \sqrt { 5 } \sec { \alpha  } =\frac { 2\csc { \alpha  }  }{ t } =-\frac { 1 }{ { t }^{ 2 } } \Rightarrow \cos { \alpha  } =-\sqrt { 5 } { t }^{ 2 }$ and $\sin { \alpha  } =-2t$

$\displaystyle \Rightarrow \cos ^{ 2 }{ \alpha  } +\sin ^{ 2 }{ \alpha  } =5{ t }^{ 4 }+4{ t }^{ 2 }=1\Rightarrow { t }^{ 2 }=\frac { 1 }{ 5 } $   

$\therefore$ (A) is true
and $\displaystyle \frac { \sin { \alpha  }  }{ \cos { \alpha  }  } =-\frac { 2t }{ -\sqrt { 5 } { t }^{ 2 } } =\frac { 2 }{ \sqrt { 5 }  } \times \frac { 1 }{ t } =\frac { 2 }{ \sqrt { 5 }  } \times \left( \pm 5 \right) $

$\therefore \tan { \alpha  } =\pm 2$

Multiple choice maths ellipse normal to an ellipse tangent and normal to an ellipse two dimensional analytical geometry-ii

lf the tangent drawn at a point $(t^{2},2t)$ on the parabola $y^{2}=4x$ is same as normal drawn at $(\sqrt{5}\cos\alpha, 2\sin\alpha)$ on the ellipse $\displaystyle \frac{x^{2}}{5}+\frac{y^{2}}{4}=1$, then which of following is not true?  

  1. $t=\displaystyle \pm\frac{1}{\sqrt{5}}$
  2. $\alpha=-\tan^{-1}2$
  3. $\alpha=\tan^{-1}2$
  4. $\alpha=\tan^{-1}4$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The tangent to y^2=4x at (t^2, 2t) is ty = x + t^2. The normal to x^2/5 + y^2/4 = 1 at (sqrt(5)cos(alpha), 2sin(alpha)) is sqrt(5)xsec(alpha) - 2ycosec(alpha) = 1. Comparing coefficients leads to the condition for the lines to be identical, which excludes option D.

Multiple choice

Which conic section is represented by the equation (y^2 = 4px)?

  1. Circle

  2. Ellipse

  3. Parabola

  4. Hyperbola
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation (y^2 = 4px) represents a parabola with vertex at the origin and axis of symmetry along the (x)-axis.

Multiple choice

What is the standard form of the equation of a parabola?

  1. \(x^2 + y^2 = r^2\)
  2. \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
  3. \(y^2 = 4px\)
  4. \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The standard form of the equation of a parabola is (y^2 = 4px), where (p) is the distance from the vertex to the focus.

Multiple choice

What is the standard form of the equation of a hyperbola?

  1. \(x^2 + y^2 = r^2\)
  2. \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
  3. \(y^2 = 4px\)
  4. \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The standard form of the equation of a hyperbola is (\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1), where (a) and (b) are the lengths of the transverse and conjugate axes, respectively.

Multiple choice

What is the equation of the directrix of a parabola with vertex at the origin and focus at ((p, 0))?

  1. \(x = -p\)
  2. \(x = p\)
  3. \(y = -p\)
  4. \(y = p\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of the directrix of a parabola with vertex at the origin and focus at ((p, 0)) is (x = -p).

Multiple choice

What is the equation of the directrix of a hyperbola with center at the origin and transverse axis (2a)?

  1. \(x = -2a\)
  2. \(x = 2a\)
  3. \(y = -2a\)
  4. \(y = 2a\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation of the directrix of a hyperbola with center at the origin and transverse axis (2a) is (x = 2a).

Multiple choice

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2).

  1. y^2 = 8x

  2. y^2 = 4x

  3. y^2 = 2x

  4. y^2 = x
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The standard equation of a parabola with vertex at the origin and focus at (0, p) is given by y^2 = 4px. Substituting p = 2, we get the equation y^2 = 4x.

Multiple choice

What is the equation of the hyperbola with center at the origin, vertices at (±3, 0), and foci at (±5, 0)?

  1. x^2 - y^2 = 9

  2. x^2 - y^2 = 25

  3. x^2 + y^2 = 9

  4. x^2 + y^2 = 25
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The standard equation of a hyperbola with center at the origin, vertices at (±a, 0), and foci at (±c, 0) is given by x^2 / a^2 - y^2 / b^2 = 1, where c^2 = a^2 + b^2. Substituting a = 3 and c = 5, we get: 5^2 = 3^2 + b^2, which gives b^2 = 16. Therefore, the equation of the hyperbola is x^2 / 9 - y^2 / 16 = 1, or equivalently, x^2 - y^2 = 25.