HARD

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On the ellipse $\frac{x^{2}}{18} + \frac{y^{2}}{8} = 1,$ then Find the point $M$ nearest to the line $2 x - 3 y + 25 = 0$ .

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Important Questions on Ellipse

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

The line

x = 8

is the directrix of the ellipse

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

with the corresponding focus

(2, 0)

. If the tangent to

E

at the point

P

in the first quadrant passes through the point

(0, 4 \sqrt{3})

and intersects the

x

-axis at

Q

, then

{(3 P Q)}^{2}

is equal to _____ .

EASY

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

The number of values of '

c

' such that the straight line

y = 4 x + c

touches the curve

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

, is

HARD

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

Let $C$ be the largest circle centred at $(2, 0)$ and inscribed in the ellipse $\frac{x^{2}}{36} + \frac{y^{2}}{16} = 1$ . If $(1, α)$ lies on $C$ , then $10 α^{2}$ is equal to ______

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

If the radius of the largest circle with centre

(2, 0)

inscribed in the ellipse

x^{2} + 4 y^{2} = 36

r

, then

12 r^{2}

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

3 x + 4 y = 12 \sqrt{2}

is a tangent o the ellipse

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

for some

a \in R,

then the distance between the foci of the ellipse is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

Let a circle of radius

4

be concentric to the ellipse

15 x^{2} + 19 y^{2} = 285

. Then the common tangents are inclined to the minor axis of the ellipse at the angle

HARD

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

Let a tangent to the curve

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

intersect the coordinate axes at the points

A

and

B

. Then, the minimum length of the line segment

A B

is ______

HARD

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

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>Interaction between Ellipse and a Line

Find the condition for the line

x \cos α + y \sin α = p

to be a tangent to the ellipse

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

HARD

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

The line

y = x + 1

meets the ellipse

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

at two points

P

and

Q

. If

r

is the radius of the circle with

P Q

as diameter then

{(3 r)}^{2}

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

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,

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

Find the equation of tangents to the ellipse

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

which is perpendicular to

x + y + 2 = 0

HARD

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

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>Interaction between Ellipse and a Line

If the maximum distance of normal to the ellipse

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

, from the origin is

1

, then the eccentricity of the ellipse is:

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

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

HARD

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

If the tangent to the parabola

y^{2} = x

at a point

(α, β), (β > 0)

is also a tangent to the ellipse,

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

then

α

is equal to:

HARD

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

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?

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

Let the ellipse

E : x^{2} + 9 y^{2} = 9

intersect the positive

x

- and

y

-axes at the points

A

and

B

respectively. Let the major axis of

E

be a diameter of the circle

C

. Let the line passing through

A

and

B

meet the circle

C

at the point

P

. If the area of the triangle with vertices

A, P

and the origin

O

\frac{m}{n}

, where

m

and

n

are coprime, then

m - n

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

Find the equation of tangents to the ellipse $2 x^{2} + y^{2} = 8$ which is parallel to $x - 2 y + 4 = 0$ .

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Interaction between Ellipse and a Line

If the line

y = 2 x + c

touches the curve

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

then

c^{2} =

On the ellipse x218+y28=1, then Find the point M nearest to the line 2 x-3 y+25=0.

Important Questions on Ellipse

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On the ellipse $\frac{x^{2}}{18} + \frac{y^{2}}{8} = 1,$ then Find the point $M$ nearest to the line $2 x - 3 y + 25 = 0$ .