Ellipse — JEE Maths Practice Questions & Solutions

hardPYQ · JEE Main 2026

Let

P(3\cos\alpha,2\sin\alpha)

,

\alpha\ne0

, be a point on the ellipse

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

,

Q

a point on the circle

x^2+y^2-14x-14y+82=0

, and

R

a point on the line

x+y=5

, such that the centroid of

\triangle PQR

is

\left(2+\cos\alpha,\ 3+\dfrac23\sin\alpha\right)

. Then the sum of the ordinates of all possible points

R

is

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easyPYQ · JEE Main 2026

Let a focus of the ellipse

E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

be

S(4,0)

and its eccentricity be

\dfrac45

. If the point

P(3,\alpha)

lies on

E

and

O

is the origin, then the area of

\triangle POS

is

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easyPYQ · JEE Main 2026

Let an ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

,

a<b

, pass through the point

(4,3)

and have eccentricity

\dfrac{\sqrt5}{3}

. Then the length of its latus rectum is

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hard

The maximum distance of the centre of the ellipse

\dfrac{x^2}{16}+\dfrac{y^2}{9}=1

from the chord of contact of mutually perpendicular tangents is:

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hard

Two tangents are drawn to the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

from the point

P(h,k)

. The points in which these tangents cut the axes are concyclic. The locus of

P

is:

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hard

Three curves are given:

C_1 : x^{2} - y^{2} = 1 \ (x < 0)

C_2 : xy = 1 \ (y > 0)

and

C_3

, an ellipse having one focus at a focus of

C_2

, the directrix corresponding to that focus being an asymptote of

C_1

, with the major axis of

C_3

of length

\dfrac83

. Find the eccentricity of

C_3

.

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hard

The line

\dfrac{x}{2}\cos\theta + \dfrac{y}{1}\sin\theta = 1

, with

\theta \in \left(\dfrac{\pi}{18},\dfrac{\pi}{15}\right)

, meets the ellipse

x^{2}+2y^{2}=6

at

P

and

Q

. Find the angle between the tangents to the ellipse at

P

and

Q

.

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hard

From every point of the line

(t+2)(x+y) = 1

,

t \ne -2

, tangents are drawn to the ellipse

4x^{2} + 16y^{2} = 1

, and the chord of contact passes through a fixed point. Find the number of integral values of

t

for which this fixed point lies inside the ellipse.

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medium

An ellipse having foci at

(3,3)

and

(-4,4)

and passing through the origin has eccentricity equal to:

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medium

The foci of an ellipse are

(-2,4)

and

(2,1)

. The point

\left(1,\dfrac{23}{6}\right)

is an extremity of the minor axis. What is the value of the eccentricity?

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medium

An ellipse with major and minor axes of length

10\sqrt3

and

10

units slides along the coordinate axes and always remains confined in the first quadrant. The locus of the centre of the ellipse is the arc of a circle. The length of this arc is:

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medium

Let

S(3,4)

and

S'(9,12)

be the foci of an ellipse and the foot of the perpendicular from

S

to a tangent to the ellipse is

(1,-4)

. Then the eccentricity of the ellipse is:

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medium

The line

lx+my+n=0

cuts the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

in points whose eccentric angles differ by

\dfrac{\pi}{2}

, if:

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medium

The normal at a variable point

P

on an ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

of eccentricity

e

meets the axes in

Q

and

R

. Then the locus of the midpoint of

QR

is a conic with eccentricity

e'

such that:

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medium

A circle touches the major axes of the two ellipses

\dfrac{x^2}{2086}+\dfrac{y^2}{2010}=1

and

\dfrac{x^2}{2010}+\dfrac{y^2}{2086}=1

and one of their common tangents, with radius

r

. Then

[r]

(greatest integer function) equals:

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medium

In the standard ellipse, the lines joining the ends of the minor axis to one focus are at right angles. The distance between the focus and the nearer vertex is

\sqrt{10}-\sqrt5

. The equation of the ellipse is:

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medium

If the tangent at a point

(a\cos\theta,b\sin\theta)

on the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

meets the auxiliary circle in two points and the chord joining them subtends a right angle at the centre, then the eccentricity is:

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medium

A focal chord through the focus

S

meets the ellipse

9x^2+16y^2=144

at

P

and

Q

.

A

and

B

are the feet of the perpendiculars from the foci

S

and

S'

to the tangent at

P

, and

O

is the centre. Then

\dfrac{SP+SQ}{SP\cdot SQ}+\dfrac{OA\cdot OB}{SA\cdot S'B}

equals:

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medium

If the ellipse

\dfrac{x^2}{4}+y^2=1

meets the ellipse

x^2+\dfrac{y^2}{a^2}=1

in four distinct points and

a=b^2-5b+7

, then

b

does not lie in:

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medium

If a variable tangent of the circle

x^2+y^2=1

intersects the ellipse

x^2+2y^2=4

at

P

and

Q

, then the locus of the point of intersection of the tangents at

P

and

Q

is:

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medium

A circle

S=0

touches the circle

x^2+y^2-4x+6y-23=0

internally and the circle

x^2+y^2-4x+8y+19=0

externally. The locus of the centre of

S=0

is a conic whose eccentricity is

k

. Then

\left[\dfrac1k\right]

(greatest integer function) is:

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medium

If the circumcentre of an equilateral triangle inscribed in

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

, with vertices having eccentric angles

\alpha,\beta,\gamma

, is

(x_1,y_1)

, then

\sum\cos\alpha\cos\beta+\sum\sin\alpha\sin\beta

is:

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medium

For an ellipse, the minor axis lies along the line

x + 2y = 3

and has length

\sqrt5

, and the foot of a directrix is

(2,3)

. Then

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medium

The ellipse

\dfrac{x^{2}}{16} + \dfrac{y^{2}}{b^{2}} = 1

is inscribed in a square whose side has length

5\sqrt2

. Find the area of the ellipse.

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medium

An ellipse has its centre at the origin, and the lines

3x + 5y = 20

and

3x - 5y = 20

touch it at the two ends of a latus rectum. Find the length of that latus rectum.

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medium

For an ellipse, one focus is

(2,3)

, the eccentricity is

\dfrac13

, and the corresponding directrix cuts the major axis at

(10,11)

. Which of the following statements are correct?

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medium

The minimum value of

f(x) = |x-a| + 2|x-2|

is

1

, and

a > 2

. Take the ellipse

\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{4} = 1

and draw tangents to it from the point

P(1,2)

. If this ellipse has eccentricity

e

, then a possible value of

e

is

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medium

A tangent

PQ

to the ellipse

\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1

cuts the positive

x

and

y

axes at

P

and

Q

respectively. The point

R

divides

PQ

internally in the ratio

2 : 1

. Given that the locus of

R

is

\dfrac{a^{2}}{x^{2}} + \dfrac{4b^{2}}{y^{2}} = \lambda

, find

\lambda

.

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medium

For how many values of

c

does the line

y = cx + c

,

c \in \mathbb{R}

, touch the curve

\dfrac{x^{2}}{4} + \dfrac{y^{2}}{1} = 1

?

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medium

Let

A

be a point on the ellipse

x^{2} + 16y^{2} = 4

, whose foci are

S_1

and

S_2

. Find the maximum value of

AS_1 \cdot AS_2

.

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easy

The equation

\dfrac{x^2}{10-a}+\dfrac{y^2}{4-a}=1

represents an ellipse if:

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easy

The eccentricity of the ellipse

(x-3)^2+(y-4)^2=\dfrac{y^2}{9}

is:

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easy

An ellipse has foci at

(9,20)

and

(49,55)

in the

xy

-plane and is tangent to the

x

-axis. The length of its major axis is:

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easy

If

\dfrac{x^2}{f(4a)}+\dfrac{y^2}{f(a^2-5)}=1

represents an ellipse with major axis as

y

-axis and

f

is a decreasing function, then

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easy

If

b^2=a^2(1-e^2)

, then for any

\alpha

,

a,b\in\mathbb{R}

,

\sqrt{a^2\cos^2\alpha+b^2\sin^2\alpha-2a^2e\cos\alpha+a^2e^2}+\sqrt{a^2\cos^2\alpha+b^2\sin^2\alpha+2a^2e\cos\alpha+a^2e^2}

is equal to:

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easy

If an ellipse slides on the rectangular coordinate axes and foci of the ellipse are at

(2,y_1)

and

(3,y_2)

, then

y_1y_2

equals

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easy

The foci of the ellipse

3x^2-4xy+3y^2=5

are

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easy

The minimum value of

\left\{(r+5-4|\cos\theta|)^2+(r-3|\sin\theta|)^2\right\}

,

\forall\,r,\theta\in\mathbb{R}

, is:

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easy

The area of the triangle inscribed in an ellipse bears the ratio

\sqrt5:3

to the area of the triangle formed by joining points on the auxiliary circle corresponding to the vertices of the first triangle. Then the eccentricity of the ellipse is:

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easy

Which one of the following is the common tangent to the ellipses

\dfrac{x^2}{a^2+b^2}+\dfrac{y^2}{b^2}=1

and

\dfrac{x^2}{a^2}+\dfrac{y^2}{a^2+b^2}=1

?

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easy

If

\tan\theta_1\tan\theta_2=-\dfrac{a^2}{b^2}

, then the chord joining the points

\theta_1

and

\theta_2

on the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

subtends a right angle at:

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easy

Let

P

be a point on the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\ (a>b)

in the first quadrant, whose foci are

S_1

and

S_2

. Then the least possible value of the circumradius of

\triangle PS_1S_2

is:

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easy

If

CF

is the perpendicular from the centre

C

of the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

on the tangent at a point

P

, and

G

is the point where the normal at

P

meets the major axis, then

CF\cdot PG

is:

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easy

Tangents are drawn to the ellipse

\dfrac{x^2}{9}+\dfrac{y^2}{5}=1

at the ends of the latus rectum. The area of the quadrilateral so formed is:

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easy

If the ellipse

\dfrac{x^2}{a^2-3}+\dfrac{y^2}{a+4}=1

is inscribed in a square of side length

a\sqrt2

units, then

a

is:

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easy

Let there exist at least one normal to an ellipse which touches a concentric circle, and the circle intersects the ellipse at four distinct points. Then the eccentricity of the ellipse is:

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easy

Tangents are drawn from any point on the circle

x^2+y^2=41

to the ellipse

\dfrac{x^2}{25}+\dfrac{y^2}{16}=1

. The angle between the two tangents is:

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easy

The conics

\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1

and

\dfrac{x^{2}}{A^{2}} - \dfrac{y^{2}}{B^{2}} = 1

, where

a > A

and

a > b

, have the same pair of foci

S_1

and

S_2

, and

P

is a point common to both. Find the quadratic equation whose roots are

S_1P

and

S_2P

.

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easy

Take the four points

\left(\pm\frac{ab}{\sqrt{a^{2}+b^{2}}},\ \pm\frac{ab}{\sqrt{a^{2}+b^{2}}}\right)

Which of the following is correct?

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Ellipse — JEE Maths Practice Questions & Solutions

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