Define the collections E 1 , E 2 , E 3 , . . . . of ellipses and R 1 , R 2 , R 3 , . . . . of rectangles as follows: E 1 : x 2 9 + y 2 4 = 1 ; R 1 : rectangle of largest area, with sides parallel to the axes, inscribed in E 1 ; E n : ellipse x 2 a n 2 + y 2 b n 2 = 1 of largest area inscribed in R n - 1 , n > 1 ; R n : rectangle of largest area, with sides parallel to the axes, inscribed in E n , n > 1 . Then which of the following options is/are correct?

∑ n = 1 N a r e a o f R n < 24 , for each positive integer N

The eccentricities of E 18 and E 19 are NOT equal

The distance of a focus from the centre in E 9 is 5 32

HARD

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If the angle between the lines joining the foci to an extremity of the minor axis of an ellipse is $90 °$ , its eccentricity is:

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Important Questions on Geometry

MEDIUM

Arithmetic>Geometry>Geometry>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

Arithmetic>Geometry>Geometry>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)

MEDIUM

Arithmetic>Geometry>Geometry>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

Arithmetic>Geometry>Geometry>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

EASY

Arithmetic>Geometry>Geometry>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

HARD

Arithmetic>Geometry>Geometry>Ellipse

Define the collections

\{E_{1}, E_{2}, E_{3}, . . . .\}

of ellipses and

\{R_{1}, R_{2}, R_{3}, . . . .\}

of rectangles as follows:

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

R_{1} :

rectangle of largest area, with sides parallel to the axes, inscribed in

E_{1};

E_{n} :

ellipse

\frac{x^{2}}{a_{n}^{2}} + \frac{y^{2}}{b_{n}^{2}} = 1

of largest area inscribed in

R_{n - 1}, n > 1;

R_{n} :

rectangle of largest area, with sides parallel to the axes, inscribed in

E_{n}, n > 1 .

Then which of the following options is/are correct?

EASY

Arithmetic>Geometry>Geometry>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

HARD

Arithmetic>Geometry>Geometry>Ellipse

e_{1}

and

e_{2}

are the eccentricities of the ellipse

\frac{x^{2}}{18} + \frac{y^{2}}{4} = 1

and the hyperbola

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

respectively and

(e_{1}, e_{2})

is a point on the ellipse

15 x^{2} + 3 y^{2} = k

, then the value of

k

is equal to

MEDIUM

Arithmetic>Geometry>Geometry>Ellipse

The eccentricity of the conic

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

EASY

Arithmetic>Geometry>Geometry>Ellipse

Eccentricity of the ellipse

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

MEDIUM

Arithmetic>Geometry>Geometry>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:

MEDIUM

Arithmetic>Geometry>Geometry>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

Arithmetic>Geometry>Geometry>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

Arithmetic>Geometry>Geometry>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

EASY

Arithmetic>Geometry>Geometry>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

EASY

Arithmetic>Geometry>Geometry>Ellipse

The major and minor axis of the ellipse

400 x^{2} + 100 y^{2} = 40000

respectively are

EASY

Arithmetic>Geometry>Geometry>Ellipse

The eccentricity of the ellipse

9 x^{2} + 25 y^{2} = 225

MEDIUM

Arithmetic>Geometry>Geometry>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:

HARD

Arithmetic>Geometry>Geometry>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:

MEDIUM

Arithmetic>Geometry>Geometry>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 :

If the angle between the lines joining the foci to an extremity of the minor axis of an ellipse is 90°, its eccentricity is:

Important Questions on Geometry

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If the angle between the lines joining the foci to an extremity of the minor axis of an ellipse is $90 °$ , its eccentricity is: