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An ellipse intersects the hyperbola $2 x^{2} - 2 y^{2} = 1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then

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Important Questions on Ellipse

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If a bar of given length moves with its extremities on two fixed straight lines at right angles, then the locus of any point on bar marked on the bar describes a/an

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

The co-ordinates of foci of the ellipse

16 x^{2} + 9 y^{2} = 144

are

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

B

and

B^{'}

are the ends of minor axis and

S

and

S^{'}

are the foci of the ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1

then the area of the rhombus

S B S^{'} B^{'}

will be

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

The equation

\frac{x^{2}}{1 - r} + \frac{y^{2}}{r - 3} + 1 = 0

represents an ellipse if

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

An ellipse, with foci at

(0,2)

and

(0, - 2)

and minor axis of length

4

, passes through which of the following points?

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If the points of intersection of the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{b^{2}} = 1$ and the circle $x^{2} + y^{2} = 4 b, b > 4$ lie on the curve $y^{2} = 3 x^{2}$ , then $b$ is equal to :

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

Let

P

be a point on the ellipse

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

and the line through

P

parallel to the

Y

-axis meets the circle

x^{2} + y^{2} = 9

Q,

where

P, Q

are on the same side of the

X

-axis. If

R

is a point on

P Q

such that

\frac{P R}{R Q} = \frac{1}{2},

then the locus of

R

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

\tan θ_{1} \times \tan θ_{2} = \frac{- a^{2}}{b^{2}}

, then the chord joining

2

points

θ_{1}

and

θ_{2}

on the ellipse

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

will subtend a right angle at

HARD

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If the point

P

on the curve,

4 x^{2} + 5 y^{2} = 20

is farthest from the point

Q (0, - 4)

, then

P Q^{2}

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If an ellipse has centre at

(0, 0),

a focus at

(- 3, 0)

and the corresponding directrix is

3 x + 25 = 0,

then it passes through the point

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If an ellipse has its foci at

(2, 0)

and

(- 2, 0)

and its length of the latus rectum is

6

, then the equation of the ellipse is

HARD

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

An ellipse inscribed in a semi-circle touches the circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. When the ellipse has the maximum possible area, its eccentricity is-

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If the area of the Ellipse is

\frac{x^{2}}{25} + \frac{y^{2}}{λ^{2}} = 1

20 π

square units, then

λ

HARD

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

The equation of the ellipse whose centre is at the origin and the

x

-axis is the major axis, which passes through the points

(- 3, 1)

and

(2, - 2)

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

A focus of an ellipse having eccentricity

\frac{1}{2}

is at

(0, 0)

and a directrix is the line

x = 4 .

Then the equation of one such ellipse is

HARD

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

Find the equation and eccentricity of the ellipse, if the centre of ellipse is same as centre of the hyperbola $4 x^{2} - 9 y^{2} + 8 x - 36 y - 68 = 0$ , length of semi major axis is $3$ units, length of minor axis is $2$ units, the equation of major axis is $x = - 1$ .

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

P

is a point on the conic

a^{2} x^{2} + b^{2} y^{2} = a^{2} (a^{2} + b^{2} - y^{2})

and

S

is a focus of that conic.

M

is the foot of the perpendicular from

P

on to a directrix of that conic nearer to

S

. If

P M = K S P

, then

K =

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

The equation of the ellipse in the standard form whose length of the latus rectum is

4

and whose distance between the foci is

4 \sqrt{2}

, is

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

Two sets

A

and

B

are as under:

A = \{(a, b) \in R \times R : |a - 5| < 1 and |b - 5| < 1\};

B = \{(a, b) \in R \times R : 4 {(a - 6)}^{2} + 9 {(b - 5)}^{2} \leq 36\} .

Then :

HARD

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

An ellipse passes through the foci of the hyperbola,

9 x^{2} - 4 y^{2} = 36

and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is

\frac{1}{2},

then which of the following points does not lie on the ellipse?

An ellipse intersects the hyperbola 2x2-2y2=1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then

Important Questions on Ellipse

Learn and Prepare for any exam you want !

An ellipse intersects the hyperbola $2 x^{2} - 2 y^{2} = 1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then