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Let $P$ be a point on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, 0 < b < a .$ Let the line parallel to $y$ -axis passing through $P$ meets the circle $x^{2} + y^{2} = a^{2}$ at the point $Q$ such that $P$ and $Q$ are on the same side of $x$ -axis. For two positive real numbers $r$ and $s$ , find the locus of the point $R$ on $P Q$ such that $P R : R Q = r : s$ , as $P$ varies over the ellipse.

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Important Questions on Ellipse

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-

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

Consider an ellipse, whose center is at the origin and its major axis is along the

x

-axis. If its eccentricity is

\frac{3}{5}

and the distance between its foci is

6

, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

In an ellipse, its foci and the ends of its major axis are equally spaced. If the length of its semi-minor axis is

2 \sqrt{2},

then the length of its semi-major axis is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If a point

P (x, y)

moves along the ellipse

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

and if

C

is the centre of the ellipse, then the sum of maximum and minimum values of

C P

HARD

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse 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

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points

(4, - 1)

and

(- 2, 2)

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 length of the latus rectum of an ellipse is

4

units and the distance between a focus and its nearest vertex on the major axis is

\frac{3}{2}

units, then its eccentricity is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

The eccentricity of the conic

x^{2} + 2 y^{2} - 2 x + 3 y + 2 = 0

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

A focus of an ellipse is at the origin. The directrix is the line

x = 4

and the eccentricity is

1 / 2 .

Then the length of the semi-major axis is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

The equation of the circle passing through the foci of the ellipse

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

, and having centre at

(0, 3)

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

Let

O (0,0)

and

A (0,1)

be two fixed points. Then, the locus of a point

P

such that the perimeter of

Δ A O P

4

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

O B

is the semi-minor axis of an ellipse,

F_{1}

and

F_{2}

are its focii and the angle between

F_{1} B

and

F_{2} B

is a right angle, then the square of the eccentricity of the ellipse is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at

(0,5 \sqrt{3}),

then the length of its latus rectum is:

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If the distance between the foci of an ellipse is

6

and the distance between its directrix is

12,

then the length of its latus rectum is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

If the line

x - 2 y = 12

is a tangent to the ellipse

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

at the point

(3, - \frac{9}{2})

, then the length of the latus rectum of the ellipse is

EASY

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

Eccentricity of the ellipse

4 x^{2} + y^{2} - 8 x + 4 y - 8 = 0

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?

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Equation of an Ellipse

Let

S

and

S^{'}

be the foci of an ellipse and

B

be any one of the extremities of its minor axis. If

Δ S^{'} B S

is a right angled triangle with right angle at

B

and area

(Δ S^{'} B S) = 8 sq . units

, then the length of a latus rectum of the ellipse is :

Important Questions on Ellipse

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