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

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A light ray emanating from the point $(- 14 \sqrt{6}, 0)$ is incident on the ellipse $25 x^{2} + 49 y^{2} = 1225$ at point $Q$ with abscicca zero, then the equation of the reflected ray after first reflection is

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Important Questions on Ellipse

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an 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

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal 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>Properties of Tangents and Normal of an Ellipse

What is the equation of the ellipse having foci

(\pm 2, 0)

and the eccentricity

\frac{1}{4} ?

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an 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 Tangents and Normal of an 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 Tangents and Normal of an 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 a point

P

on the ellipse, then

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

is equal to

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

The normal at a point

P

on the ellipse

x^{2} + 4 y^{2} = 16

meets the

x

-axis at

Q

. If

M

is the mid-point of the line segment

P and Q

, then the locus of

M

intersects the latus rectum of the given ellipse at the points.

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

A line

2 x + 3 y = 4

is reflected from the inner surface of ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1

. The reflected line will pass through

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

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

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an 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 Tangents and Normal of an Ellipse

An ellipse slides between two lines at right angle to one another. The locus of its centre is

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

F_{1}

F_{2}

are the foci of ellipse

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

, the product of perpendicular from

F_{1}

F_{2}

on the tangent at point

P (3 \cos θ, 2 \sin θ)

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an 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

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

The centre of the ellipse

14 x^{2} - 4 x y + 11 y^{2} - 44 x - 58 y + 71 = 0

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

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

x = 4

and the eccentricity is

\frac{1}{2,}

then length of semi major axis is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

P S Q

is a focal chord of the ellipse

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

a > b,

then the harmonic mean of

S P

and

S Q

HARD

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an 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

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

F_{1}

and

F_{2}

are the foot of the perpendiculars from the foci

S_{1}

and

S_{2}

of an ellipse

\frac{x^{2}}{5} + \frac{y^{2}}{3} = 1

on the tangent at any point

P

on the ellipse, then

(S_{1} F_{1}) (S_{2} F_{2})

is equal to

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

An ellipse with major and minor axes length as

2 a

and

2 b

touches coordinate axes in first quadrant and having foci

(x_{1}, y_{1})

and

(x_{2}, y_{2})

then the value of

x_{1} x_{2}

and

y_{1} y_{2}

EASY

Mathematics>Coordinate Geometry>Ellipse>Properties of Tangents and Normal of an Ellipse

Cotyledons are also called-

A light ray emanating from the point -146,0 is incident on the ellipse 25x2+49y2=1225 at point Q with abscicca zero, then the equation of the reflected ray after first reflection is

Important Questions on Ellipse

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A light ray emanating from the point $(- 14 \sqrt{6}, 0)$ is incident on the ellipse $25 x^{2} + 49 y^{2} = 1225$ at point $Q$ with abscicca zero, then the equation of the reflected ray after first reflection is