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Equations of the common tangent of the ellipse $\frac{(x + 1)^{2}}{2^{2}} + \frac{(y - 1)^{2}}{3^{2}} = 1$ and the circle $(x + 1)^{2} + (y - 1)^{2} = 4$ are

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Important Questions on Ellipse

EASY

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

On the ellipse

\frac{x^{2}}{8} + \frac{y^{2}}{4} = 1

, let

P

be a point in the second quadrant such that the tangent at

P

to the ellipse is perpendicular to the line

x + 2 y = 0

. Let

S

and

S^{'}

be the foci of the ellipse and

e

be its eccentricity. If

A

is the area of the triangle

{SPS}^{'}

, then the value of

(5 - e^{2}) \cdot A

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus ractum to the ellipse

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

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

If a tangent to the ellipse

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

meets the tangents at the extremities of its major axis at

B

and

C,

then the circle with

B C

as diameter passes through the point.

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

Find the equation of tangent and normal to the ellipse

x^{2} + 8 y^{2} = 33

(- 1, 2)

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

Let the ellipse,

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

pass through the point

(2, 3)

and have eccentricity equal to

\frac{1}{2} .

Then equation of the normal to this ellipse at

(2, 3)

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

If a tangent to the ellipse

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

meets its major axis and minor axis at

M

and

N

respectively, then prove that

\frac{a^{2}}{(C M)^{2}} + \frac{b^{2}}{(C N)^{2}} = 1

, where

C

is the centre of the ellipse.

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

Let

E

be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at

(3, - 4),

one focus at

(4, - 4)

and one vertex at

(5, - 4) .

m x - y = 4, m > 0

is a tangent to the ellipse

E,

then the value of

5 m^{2}

is equal to _____ .

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

The locus of the foot of perpendicular drawn from the centre of the ellipse

x^{2} + 3 y^{2} = 6

on any tangent to it is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

If the normal drawn at one end of the latus rectum of the ellipse

b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2}

with eccentricity '

e

' passes through one end of the minor axis, then

EASY

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

Consider a tangent to the ellipse

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

at any point. The locus of the mid-point of the portion intercepted between the axes is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

Find the equations of tangents to the ellipse

2 x^{2} + 3 y^{2} = 11

at the points whose ordinate is

1

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

Let

x = 4

be a directrix to an ellipse whose centre is at the origin and its eccentricity is

\frac{1}{2} .

P (1, β), β > 0

is a point on this ellipse, then the equation of the normal to it at

P

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

The value of

λ

for which the line

y = x + λ

touches the ellipse

9 x^{2} + 16 y^{2} = 144

is/are

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

If a circle

(x - 1)^{2} + y^{2} = r^{2}

touches the ellipse

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

internally, then

r =

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

The eccentricity of an ellipse whose centre is at the origin is

\frac{1}{2}

. If one of its directrices is

x = - 4

, then the equation of the normal to it at

(1, \frac{3}{2})

is:

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

The tangent to a suitable conic (Column 1) at

(\sqrt{3}, \frac{1}{2})

is found to be

\sqrt{3} x + 2 y = 4

, then which of the following options is the only Correct combination?

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

If the tangent at a point on the ellipse

\frac{x^{2}}{27} + \frac{y^{2}}{3} = 1

meets the coordinate axes at

A

and

B

, and

O

is the origin, then the minimum area (in sq. units) of the triangle

O A B

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

The values of

λ

for which the line

y = x + λ

touches the ellipse

9 x^{2} + 16 y^{2} = 144,

are

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity

e

of the ellipse satisfies :

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to an Ellipse

Let the line

y = m x

and the ellipse

2 x^{2} + y^{2} = 1

intersect at a point

P

in the first quadrant. If the normal to this ellipse at

P

meets the co-ordinate axes at

(- \frac{1}{3 \sqrt{2}}, 0)

and

(0, β)

, then

β

is equal to

Equations of the common tangent of the ellipse (x+1)222+(y-1)232=1 and the circle (x+1)2+(y-1)2=4 are

Important Questions on Ellipse

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Equations of the common tangent of the ellipse $\frac{(x + 1)^{2}}{2^{2}} + \frac{(y - 1)^{2}}{3^{2}} = 1$ and the circle $(x + 1)^{2} + (y - 1)^{2} = 4$ are