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

Tangents are drawn to the ellipse x 2 a 2 + y 2 b 2 = 1 from the point a 2 a 2 - b 2 , a 2 + b 2 prove that they intersect the ordinate through the nearer focus a distance equal to the major axis.

Equation to the ellipse is: x 2 a 2 + y 2 b 2 = 1 Coordinates of the point are: a 2 a 2 - b 2 , a 2 + b 2 As we know that, b 2 = a 2 1 - e 2 ⇒ a e , a 2 + b 2 Equation to the pair of tangents will be given by: S S 1 = T 2 , where S = x 2 a 2 + y 2 b 2 - 1 S 1 = a 2 e 2 a 2 + a 2 + b 2 b 2 - 1 = 1 e 2 + a 2 b 2 = 1 e 2 + 1 1 - e 2 = 1 e 2 1 - e 2 And T = x a e + y a 2 + b 2 b 2 - 1 So, equation to the pair of tangents will be: 1 e 2 1 - e 2 x 2 a 2 + y 2 b 2 - 1 = x a e + y a 2 + b 2 b 2 - 1 2 These tangents meets the latus rectum(x=ae) So, putting the value of x we get: ⇒ 1 - e 2 + y 2 e 2 b 2 ( 1 - e 2 ) = y 2 a 2 + b 2 b 4 And as we know b 2 = a 2 1 - e 2 ⇒ 1 - e 2 + y 2 a 2 e 2 1 - e 2 2 = y 2 2 a 2 - a 2 e 2 a 4 1 - e 2 2 ⇒ y 2 1 a 2 e 2 1 - e 2 2 - 2 - e 2 a 2 1 - e 2 2 = 1 e 2 ⇒ y 2 1 - 2 e 2 + e 4 a 2 e 2 1 - e 2 2 = 1 e 2 ⇒ y 2 1 - e 2 2 a 2 e 2 1 - e 2 2 = 1 e 2 ⇒ y 2 = a 2 Hence, y = ± a Let y 1 and y 2 be the ordinates of the points then y 1 - y 2 = a - - a = 2 a = major axis. Hence Proved.

Identify correct statement(s) about conic x – 5 2 + y – 7 2 + x + 1 2 + y + 1 2 = 12

ellipse with major axis 4 x – 3 y + 1 = 0

hyperbola with foci 5 , 7 and – 1 , – 1

HARD

Earn 100

An equilateral triangle is inscribed in an ellipse whose equation is $x^{2} + 4 y^{2} = 4$ , one vertex of triangle is $(0, 1)$ and if one altitude is contained in $y$ -axis and length of each side is $\sqrt{\frac{p}{q}}$ (where $p$ and $q$ are relatively prime), then the value of $\frac{p - 3 q}{9} =$

50% studentsanswered this correctly

Important Questions on Ellipse

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of 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>Ellipse>Properties of Ellipse

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse,

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

from any of its foci?

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of 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>Properties of Ellipse

What is the equation of the ellipse having foci

(\pm 2, 0)

and the eccentricity

\frac{1}{4} ?

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

Consider the curve

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

. The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

If a line

x + y - 2 = 0

is tangent to an ellipse at a point

(α, β)

with foci

S_{1} (3, 2)

and

S_{2} (4, 7)

, then the value of

α + β

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

The area of the quadrilateral formed by the tangents at the end points of latus-rectum to the ellipse

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

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

A tangent at any point

P

on the ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{8} = 1

intersect the major axis at point

R

and

M, M^{'}

are foot of perpendiculars drawn from the focus

S

and

S^{'}

of the ellipse if

RS : {RS}^{'} = 2 : 3 .

Then

SM

is equal to

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

The angle between the pair of tangents drawn to the ellipse

3 x^{2} + 2 y^{2} = 5

from the point

(1, 2)

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

For an ellipse, prove that the straight lines, joining each focus to the foot of the perpendicular from the other focus upon the tangent at any point

P

, meet on the normal

P G

and bisect it.

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

An ellipse slides between two perpendicular straight lines. Then the locus of its centre is-

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

An ellipse has the point

(1, - 1)

and

(2, - 1)

as its foci and

x + y = 5

as one of its tangent, then the value of

a^{2} + b^{2}

, where

a, b

are the length of semi-major and semi-minor axis of ellipse respectively, is

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

M_{1}

and

M_{2}

are the feet of the perpendiculars from the foci

S_{1}

and

S_{2}

of the ellipse

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

on the tangent at any point

P

on the ellipse, then

(S_{1} M_{1}) (S_{2} M_{2})

is equal to

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

A tangent at any point

P

on the ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{8} = 1

intersect the major axis at point

R

and

M, M^{'}

are foot of perpendiculars drawn from the focus

S

and

S^{'}

of the ellipse. If

R S : R S^{'} = 2 : 3

, then

S M

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

Tangents are drawn to the ellipse

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

from the point

(\frac{a^{2}}{\sqrt{a^{2} - b^{2}}}, \sqrt{a^{2} + b^{2}})

prove that they intersect the ordinate through the nearer focus a distance equal to the major axis.

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

From the focus

(- 5, 0)

of the ellipse

\frac{x^{2}}{45} + \frac{y^{2}}{20} = 1

, a ray of light is sent which makes an angle

\cos^{- 1} (\frac{- 1}{\sqrt{5}})

with the positive direction of

x

-axis, upon reaching the ellipse surface, the ray is reflected from it. Slope of the reflected ray is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

An ellipse has

O B

as semi minor axis,

F

and

F

it's foci and the angle

F B F'

is a right angle. Then, the eccentricity of the ellipse is

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

If the locus of reflection of the ellipse

\frac{{(x - 4)}^{2}}{16} + \frac{{(y - 3)}^{2}}{9} = 1

about the line

x - y - 2 = 0

16 x^{2} + 9 y^{2} + k_{1} x - 36 y + k_{2} = 0,

then, find

\frac{k_{1} + k_{2}}{220}

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

The locus of feet of perpendicular from either foci of the ellipse $(x - y + 1)^{2} + (2 x + 2 y - 6)^{2} = 20$ on any tangent is:

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Ellipse

Identify correct statement(s) about conic

\sqrt{{(x - 5)}^{2} + {(y - 7)}^{2}} + \sqrt{{(x + 1)}^{2} + {(y + 1)}^{2}} = 12

An equilateral triangle is inscribed in an ellipse whose equation is x2+4y2=4, one vertex of triangle is (0,1) and if one altitude is contained in y-axis and length of each side is pq (where p and q are relatively prime), then the value of p-3q9=

Important Questions on Ellipse

Learn and Prepare for any exam you want !