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

MEDIUM

Earn 100

Let an ellipse with centre $(1, 0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along x-axis. If its minor axis subtends an angle $60^{\circ}$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.

80% studentsanswered this correctly

Important Questions on Conic Sections

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>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>Conic Sections>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>Conic Sections>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

Mathematics>Coordinate Geometry>Conic Sections>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?

HARD

Mathematics>Coordinate Geometry>Conic Sections>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>Conic Sections>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

HARD

Mathematics>Coordinate Geometry>Conic Sections>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>Conic Sections>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

Mathematics>Coordinate Geometry>Conic Sections>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>Conic Sections>Ellipse

The eccentricity of the conic

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

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>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>Conic Sections>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 :

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>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

Mathematics>Coordinate Geometry>Conic Sections>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

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>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>Conic Sections>Ellipse

The major and minor axis of the ellipse

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

respectively are

HARD

Mathematics>Coordinate Geometry>Conic Sections>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

EASY

Mathematics>Coordinate Geometry>Conic Sections>Ellipse

Eccentricity of the ellipse

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

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>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)

EASY

Mathematics>Coordinate Geometry>Conic Sections>Ellipse

The eccentricity of the ellipse whose major axis is three times the minor axis is

Let an ellipse with centre 1, 0 and latus rectum of length 12 have its major axis along x-axis. If its minor axis subtends an angle 60∘ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.

Important Questions on Conic Sections

Learn and Prepare for any exam you want !

Let an ellipse with centre $(1, 0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along x-axis. If its minor axis subtends an angle $60^{\circ}$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.