Mathematics

Conic Sections

285 Questions

Conic sections deal with the geometry of curves like ellipses, hyperbolas, and parabolas formed by the intersection of a plane with a cone. This is an important topic in advanced mathematics sections of competitive exams. Practice these questions to understand eccentricity, focal distances, and equations of tangents and normals.

Ellipse equationsHyperbola eccentricityFocal distance calculationsTangent and normal equationsConic standard forms

Conic Sections Questions

Multiple choice position of point wrt ellipse ellipse maths

Let $5x^2+7y^2=140$, then $(3,-4)$ is:

  1. Outside the ellipse

  2. Inside the ellipse

  3. On the ellipse

  4. Data insufficient
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equation of ellipse :   ${ 5x }^{ 2 }+{ 7y }^{ 2 }=140$

putting point $(3,-4)$ in equation of ellipse
     $=5\times { 3 }^{ 2 }+7{ \left( -4 \right)  }^{ 2 }-140$
     $=45+112-140$
     $=17(>0)$
Hence, $(3,-4)$ is outside the ellipse.

Multiple choice position of point wrt ellipse ellipse maths

Let $5x^2+7y^2=140$, then Position of $(4,-3)$ relative to the ellipse is

  1. Inside the ellipse.

  2. Outside the ellipse.

  3. On the ellipse.

  4. None of the above.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Equation of ellipse : ${ 5x }^{ 2 }+{ 7y }^{ 2 }=140$

putting point (4,3) in equation
     $=5\times { \left( 4 \right)  }^{ 2 }+7{ \left( -3 \right)  }^{ 2 }-140$
     $=80+63-140$
     $=3\left( >0 \right) $
Hence, point is outside the ellipse.

Multiple choice position of point wrt ellipse ellipse maths

Let $\dfrac {(x-3) ^2}9+\dfrac {(y-4) ^2}{16}=1$ then   $(3,4)$ is 

  1. Inside the ellipse

  2. Outside the ellipse

  3. On the ellipse

  4. Centre of the ellipse
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

If equation of ellipse is $\dfrac { { \left( x-h \right)  }^{ 2 } }{ { a }^{ 2 } } +\dfrac { { \left( y-k \right)  }^{ 2 } }{ { b }^{ 2 } } =1$,  centre $\left( h,k \right) $

for ellipse   $\dfrac { { \left( x-3 \right)  }^{ 2 } }{ 9 } +\dfrac { { \left( y-4 \right)  }^{ 2 } }{ 16 } =1$
            Centre of ellipse (3,4)
and centre is inside ellipse only.

Multiple choice position of point wrt ellipse ellipse maths

Let $5x^2+7y^2=140$, then $(0,0)$ is: 

  1. Inside the ellipse

  2. On the ellipse

  3. Outside the ellipse

  4. Centre of the ellipse
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

Equation of ellipse    ${ 5x }^{ 2 }+{ 7y }^{ 2 }=140$

   Dividing both sides by $140$
           $\dfrac { { x }^{ 2 } }{ 28 } +\dfrac { { y }^{ 2 } }{ 20 } =1$
Comparing with $\dfrac { { \left( x-h \right)  }^{ 2 } }{ { a }^{ 2 } } +\dfrac { { \left( y-k \right)  }^{ 2 } }{ { b }^{ 2 } } =1$
          $(h,k)\ =\ (0,0)$
Hence (0,0) is centre of given ellipse and it inside the ellipse.

Multiple choice position of point wrt ellipse ellipse maths

Let $5x^2+7y^2=140$, then $(\sqrt {14},\sqrt {10})$ is:

  1. Outside the ellipse

  2. Inside the ellipse

  3. On the ellipse

  4. Centre of the ellipse
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

          ${ 5x }^{ 2 }+{ 7y }^{ 2 }=140$          (equation of ellipse)

putting $\left( \sqrt { 14 } ,\sqrt { 10 }  \right) $ in equation
            $=5\times { \left( \sqrt { 14 }  \right)  }^{ 2 }+7\times { \left( \sqrt { 10 }  \right)  }^{ 2 }-140$
            $=70+70-140$
            $=0$
Hence, point lie on the ellipse.

Multiple choice position of point wrt ellipse ellipse maths

The position of the point $(1, 2)$ relative to the ellipse $2x^{2} + 7y^{2} = 20$ is

  1. outside the ellipse

  2. inside the ellipse but not at the focus

  3. on the ellipse

  4. at the focus
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$f\left( x,y \right) ={ 2x }^{ 2 }+7{ y }^{ 2 }-20$

Point will be outside of ellipse if $f\left( 1,2 \right) >0$

on ellipse if $f\left( 1,2 \right) =0$

will be inside if $f\left( 1,2 \right) <0$

$ f\left( 1,2 \right) ={ 2\times 1 }^{ 2 }+7{ \times 2 }^{ 2 }-20=10$

Here we see that $f\left( 1,2 \right) >0$ so point will be outside of ellipse

So correct answer will be option A

Multiple choice position of point wrt ellipse ellipse maths

If $a$ and $c$ positive real number and the ellipse $\dfrac { { x }^{ 2 } }{ { 4c }^{ 2 } } +\dfrac { { y }^{ 2 } }{ { c }^{ 2 } } =1$ has four distinet points in common with the circle ${ x }^{ 2 }+{ y }^{ 2 }=9{ a }^{ 2 }$, then

  1. $6ac+9{ a }^{ 2 }-2{ c }^{ 2 }>0$
  2. $6ac+9{ a }^{ 2 }-2{ c }^{ 2 }<0$
  3. $9ac-9{ a }^{ 2 }-2{ c }^{ 2 }<0$
  4. $9ac-9{ a }^{ 2 }-2{ c }^{ 2 }>0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The ellipse is x^2/(4c^2) + y^2/c^2 = 1. The circle is x^2 + y^2 = 9a^2. For four distinct intersection points, the circle must be larger than the minor axis but smaller than the major axis of the ellipse. This leads to the inequality 9ac - 9a^2 - 2c^2 < 0.

Multiple choice position of point wrt ellipse ellipse maths

An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is $2/3$ then the eccentricity of the ellipse is:

  1. $\dfrac{2\sqrt{2}}{3}$
  2. $\dfrac{\sqrt{5}}{3}$
  3. $\dfrac{8}{9}$
  4. $\dfrac{2}{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The area of the circle is pi*a^2. The area of the ellipse is pi*a*b. Probability of being outside = 1 - (Area_ellipse/Area_circle) = 1 - b/a = 2/3. Thus b/a = 1/3. Since b^2 = a^2(1-e^2), we have (b/a)^2 = 1-e^2. So 1/9 = 1-e^2, e^2 = 8/9, e = sqrt(8)/3 = 2sqrt(2)/3.

Multiple choice position of point wrt ellipse ellipse maths

The segment of the tangent at the point P to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, intercepted by the auxiliary circle subtends a right angle at the origin. If the eccentricity of the ellipse is smallest possible, then the point P can be

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

Equation of tangent at $P (a cos \theta, b sin \theta)$ is
$\frac{x \cos \theta}{a}+\frac{y \sin \theta}{b}=1$
$\therefore$ Equation of the lines OA and OB is
$x^{2}+y^{2}-a^{2}\left(\frac{x \cos \theta}{a}+\frac{y \sin \theta}{b}\right)^{2}=0$
since the lines OA and OB are perpendicular
$\therefore$ $1- \cos ^{2}\theta + 1 - \frac{a^{2}}{b^{2}} \sin^{2}\theta=0$
i.e $\sin^{2}\theta + 1-\frac{a^{2}}{b^{2}} \sin^{2}\theta$
i.e $\frac{b^{2}}{a^{2}}=\frac{sin^{2}\theta}{1+ \sin^{2} \theta}$
$\therefore e^{2}=1-\frac{b^{2}}{a^{2}}=1-\frac{\sin^{2} \theta}{1+\sin^{2}\theta}=\frac{1}{1+\sin^{2}\theta}$
$\therefore$ e is least if $\theta = \pm \frac{\pi}{2}$
$\therefore$ the point P is $(0,\pm b)$

Multiple choice position of point wrt ellipse ellipse maths

If one end of the diameter of the ellipse $4x^2+y^2=16$ is $(\sqrt 3, 2)$, then the other end is:

  1. $(\sqrt 3, 2)$
  2. $(-\sqrt 3, 2)$
  3. $(-\sqrt 3, -2)$
  4. $(\sqrt 3, -2)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The center of ellipse is $(0,0)$

The diameter of ellipse will pass through the center of ellipse and center will divide the diameter into two halfs
One end of diameter is $(\sqrt3,2) $ , Let the other end be $(h,k)$
We have $h+\sqrt3=0$ and $k+2=0$
$\Rightarrow h=-\sqrt3,k=-2$
Therefore option $C$ is correct

Multiple choice position of point wrt ellipse ellipse maths

The point P on the ellipse $4x^2+9y^2=36$ is such that the area of the $\Delta PF _1F _2=\sqrt{10} Sq$ units, where $F _1.F _2$ are Foci. Then P has the coordinates

  1. $(\pm\dfrac{3}{\sqrt{2}},\sqrt{2})$
  2. $(\dfrac{3}{2},2)$
  3. $(\dfrac{-3}{2},-2)$

  4. NONE
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$4x^2+9y^2=36\Rightarrow \dfrac {x^2}9+\dfrac {y^2}4=1$

So, $a=3,b=2$

In an ellipse, Focal points, $F=(\pm\sqrt {a^2-b^2},0)=(\pm\sqrt 5,0)$
So, $F _1F _2=2\sqrt 5$ which is the base of the triangle $PF _1F _2$

An arbitrry point P on ellipse is $(acos\theta,b\sin\theta)$
So, $Height\ of\ \triangle PF _1F _2=b\sin\theta=2\sin\theta$

ie, $Area=\dfrac 12base\times height=\dfrac 12\times 2\sqrt 5\times2\sin\theta=2\sqrt 5\sin\theta=\sqrt {10}$
$\Rightarrow \sin\theta=\dfrac 1{\sqrt 2}$
$\Rightarrow \cos\theta=\pm\dfrac 1{\sqrt 2}$
And $P=(a\cos\theta,b\sin\theta)=(\pm\dfrac 3{\sqrt 2},\sqrt 2)$

So, Option$ A$ has one of the anwers.

Multiple choice position of point wrt ellipse ellipse maths

Which of the following is an (x,y) coordinate pair located on the ellipse $4x^2 + 9y^2 = 100$?

  1. $(1, 3.5)$
  2. $(1.4, 3.2)$
  3. $(1.9, 2.9)$
  4. $(2.3, 3.1)$
  5. $(2.7, 2.6)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
  • Substitute each and every point on given ellipse and see which will satisfy the equation
  • Take point $(1.4,3.2)$ , when we substitute we get $4(1.96)+9(10.24) = 100$
  • Therefore correct answer is option $B$
Multiple choice

The eccentricity of an elliptical orbit is a measure of how much the orbit deviates from a circle.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The eccentricity of an elliptical orbit is a measure of how much the orbit deviates from a circle, with a value between 0 and 1.

Multiple choice

What is the parameter that describes the orientation of the ellipse in the orbital plane?

  1. Eccentricity

  2. Semi-major axis

  3. Inclination

  4. Argument of periapsis
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Argument of periapsis is the angle between the ascending node and the periapsis of an orbit.

Multiple choice

What is the equation of an ellipsoid?

  1. $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$
  2. $x^2 + y^2 + z^2 = a^2$
  3. $x^2/a^2 + y^2/b^2 - z^2/c^2 = 1$
  4. $x^2 + y^2 - z^2 = a^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of an ellipsoid is $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$, where $a$, $b$, and $c$ are the lengths of the semi-major axis, semi-minor axis, and semi-vertical axis, respectively.