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 mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

The equation $y^2+3 =2( 2x +y)$ represents a parabola with vertex at 

  1. $\left(\dfrac{1}{2}, 1\right) $ and axis parallel to $y$-axis
  2. $\left(\dfrac{1}{2}, 1\right) $ and axis parallel to $ x$-axis
  3. $\left(\dfrac{1}{2}, 1\right) $ and focus at $\left(\dfrac{3}{2}, 1\right)$
  4. $\left(1, \dfrac{1}{2},\right) $ and focus at $\left(\dfrac{3}{2}, 1\right)$
Reveal answer Fill a bubble to check yourself
B,C Correct answer
Explanation

$y^2+3=2(2x+y)$ represents parabola.


$y^2+3=4x+2y$


$y^2-2y+3=4x$

$y^2-2y+1+3=4x+1$

$(y-1)^2=4x-2$

$(y-1)^2=4(x-\dfrac{1}{2})$

So, the vertex of parabola$=\left(\dfrac{1}{2},1\right)$ and axis is parallel to x axis.

$a=1$

Focus$=\left(\dfrac{1}{2}+1,1\right)$

               $=\left(\dfrac{3}{2},1\right)$

Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

If the equation of parabola is ${x}^{2}=-9y$, then the equation of the directrix and the length of latus rectum are 

  1. $y=-\dfrac {9}{4}, 8$
  2. $x=\dfrac {-9}{4}, 9$
  3. $y=\dfrac {9}{4}, 9$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
The given equation is of the form ${x}^{2} = - 4ay$, where a is positive.
Therefore, the focus is on the y-axis in the negative direction and parabola opens downwards.
Therefore,
Given equation of parabola-
${x}^{2} = -9 y ..... \left( 1 \right)$
Standard equation of parabola-
${x}^{2} = -4ay ..... \left( 2 \right)$
Comparing ${eq}^{n} \left( 1 \right) \& \left( 2 \right)$, we have
$a = \cfrac{9}{4}$
As we know that, for parabola in the form ${x}^{2} = -4ay$, equation of directrix for parabola is-
$y = a$
$\therefore$ For ${x}^{2} = -9y$, equation of directrix is-
$y = \cfrac{9}{4}$
Length of latus rectum, $l = 4a$
As $a = \cfrac{9}{4}$,
$\therefore \; l = 4 \times \cfrac{9}{4} = 9$
Hence, the equation of directrix will be $y = \cfrac{9}{4}$ and the lngth of latus rectum will be 9.
Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

If $\displaystyle \left ( 2,0 \right )$ is the vertex and $y -$ axis the directrix of a parabola,find the coordinates of focus. 

  1. Focus is $\displaystyle \left ( 2,0 \right )$
  2. Focus is $\displaystyle \left ( 4,0 \right )$
  3. Focus is $\displaystyle \left ( 8,0 \right )$
  4. Focus is $\displaystyle \left ( -4,0 \right )$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Distance of vertex from directrix $y -$ axis is $2$ and we know it
is half the distance of focus from directrix.
$\displaystyle \therefore $ Focus is $\displaystyle \left ( 4,0 \right ).$

Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

The focal distance of a point $P$ on the parabola $y^2=12x$ if the ordinate of $P$ is $6$, is

  1. $12$
  2. $6$
  3. $3$
  4. $9$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given parabola is $y^2=12x$    ....$(i)$
Here $a=3$
For point $P(x, y), y=6$
This point lie on the parabola 
$\therefore (6)^2=12x\Rightarrow x=3$
Now, focal distance of point $P$ is $x+a=6$

Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

Find the equation of the parabola whose focus is $S(3,5)$ and vertex is $A(1,3)$.

  1. $\begin{array}{}\\ \Rightarrow \left| \right| = {\left( {x + y} \right)^2} = 2\left[ {{{\left( {x - 3} \right)}^2} + {{\left( {y - 5} \right)}^2}} \right]\end{array}$
  2. $\begin{array}{}\\ \Rightarrow \left| \right| = {\left( {x + y} \right)^2} = 2\left[ {{{\left( {x - 6} \right)}^2} + {{\left( {y - 6} \right)}^2}} \right]\end{array}$<span class="mrow"><span class="mtable">
  3. $\begin{array}{}\\ \Rightarrow \left| \right| = {\left( {x + y} \right)^2} = 2\left[ {{{\left( {x - 11} \right)}^2} + {{\left( {y - 11} \right)}^2}} \right]\end{array}$
  4. $\begin{array}{}\\ \Rightarrow \left| \right| = {\left( {x + y} \right)^2} = 2\left[ {{{\left( {x - 7} \right)}^2} + {{\left( {y - 7} \right)}^2}} \right]\end{array}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Slope of axis$=\frac{{5 - 3}}{{3 - 1}} = \frac{2}{2} = 1$
Slope of directrix$=-1$
equation of tangent at vertex $A$
$\begin{array}{l}Pt(1,3),m =  - 1\ \Rightarrow y - 3 =  - 1\left( {x - 1} \right)\ \Rightarrow x + y - 4 = 0\end{array}$
equation of directrix
$\begin{array}{l}x + y = \lambda \a = SA\ = \sqrt {4 + 4}  = 2\sqrt 2 \end{array}$
$A$ is midpoint of $PS$
$\begin{array}{l}\frac{{n + 3}}{2} = 1,\frac{{k + 5}}{2} = 3\ \Rightarrow n =  - 1,k = 1\end{array}$
$(-1,1)$ lies on directrix
$-1+1=\lambda=0$
equation of diectrix: $L:y+x=0$
$\begin{array}{l}QO = QS\ \Rightarrow \left| {\frac{{l + m}}{{\sqrt 2 }}} \right| = \sqrt {{{\left( {l - 3} \right)}^2} + {{\left( {m - 5} \right)}^2}} \ \Rightarrow {\left( {l + m} \right)^2} = 2\left[ {{{\left( {l - 3} \right)}^2} + {{\left( {m - 5} \right)}^2}} \right]\ \Rightarrow {\left( {x + y} \right)^2} = 2\left[ {{{\left( {x - 3} \right)}^2} + {{\left( {y - 5} \right)}^2}} \right]\end{array}$

Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

The equation $(13x - 1)^{2} + (13y - 1)^{2} = k(5x - 12y + 1)^{2}$ will represent a parabola if

  1. $k = 2$
  2. $k = 81$
  3. $k = 169$
  4. $k = 1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
We know that the general equation of the ellipse is 
${\left(x-h\right)}^{2}+{\left(y-k\right)}^{2}=e\dfrac{{\left(lh+mk+n\right)}^{2}}{{l}^{2}+{m}^{2}}$        ................$(1)$

${\left(13x-1\right)}^{2}+{\left(13y-1\right)}^{2}=k{\left(5x-12y+1\right)}^{2}$

$\Rightarrow\,{13}^{2}{\left(x-\dfrac{1}{13}\right)}^{2}+{13}^{2}{\left(y-\dfrac{1}{13}\right)}^{2}=k{\left(5x-12y+1\right)}^{2}$

$\Rightarrow\,169\left[{\left(x-\dfrac{1}{13}\right)}^{2}+{\left(y-\dfrac{1}{13}\right)}^{2}\right]=k{\left(5x-12y+1\right)}^{2}$

$\Rightarrow\,{\left(x-\dfrac{1}{13}\right)}^{2}+{\left(y-\dfrac{1}{13}\right)}^{2}=\dfrac{k}{169}{\left(5x-12y+1\right)}^{2}$       ...........$(2)$


A parabola has its eccentricity $e=1$

Comparing equations $(1)$ and $(2)$ we get

$\Rightarrow\,k=1$

$\therefore\,k=1$

Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

The focus of the parabola $y ^ { 2 } = 4 y - 4 x$ is

  1. $( 0,2 )$
  2. $( 1,2 )$
  3. $( 4,2 )$
  4. $( 1,3 )$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\Rightarrow y^2 = 4y - 4x$

$\Rightarrow y^2 - 4y = -4x$
$\Rightarrow y^2 - 4y + 4 = -4x + y$
$\Rightarrow (y - 2)^2 = -4 (x - 1)$
vertex = $(1, 2)$
$\dfrac{1}{4 p} = \dfrac{-1}{4} \rightarrow p = -1$
$\therefore$ Focus = $(-1 + 1, 2 + 0) = (0, 2)$

Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

The equation of the directrix of the parabola, $y ^ { 2 } + 4 y + 4 x + 2 = 0$ is -

  1. $x = - 1$
  2. $x = 1$
  3. $x = - \dfrac {3 }{ 2}$
  4. $x = \dfrac {3 }{ 2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given equation of parabola
$y^2+4y+4x+2=0$
or $(y+2)^2+4x+2=0$
or $(y+2)^2=-4(x-12)$
Its directrix  is x−12x−12=1=1
or x=32
Hence D is the correct answer.

y2+4y+4x+2=