MEDIUM

Earn 100

If a number of ellipses whose major axis is $x$ -axis and the minor axis is $y$ -axis be described having the same length of the major axis as $2$ but a variable minor axis, then the tangents at the ends of their latus rectum pass through fixed points whose distance from the centre is equal to

100% studentsanswered this correctly

Important Questions on Ellipse

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 tangent and normal to the ellipse

3 x^{2} + 5 y^{2} = 32

at the point

P (2, 2)

meet the

x

-axis at

Q

and

R

, respectively. Then the area (in sq. units) of the triangle

P Q R

is:

MEDIUM

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

The length of the minor axis (along

y

-axis) of an ellipse in the standard form is

\frac{4}{\sqrt{3}}

. If this ellipse touches the line

x + 6 y = 8

then its eccentricity is:

MEDIUM

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

p

and

p^{'}

denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length

a

, respectively, on a tangent to the ellipse and

r

denotes the focal distance of the point, then

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

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)

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

HARD

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

The locus of the midpoints of the portion of the tangents of the ellipse

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

intercepted between the coordinate axes 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 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>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

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:

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

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 :

MEDIUM

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

A variable tangent to the ellipse

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

makes intercepts on both the axes. The locus of the middle point of the portion of the tangent between the coordinate axes is

MEDIUM

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

The points of intersection of the perpendicular tangents drawn to the ellipse

4 x^{2} + 9 y^{2} = 36

lie on the curve

HARD

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

If tangents are drawn to the ellipse

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

at the ends of latus recta, then the area of the quadrilateral thus formed is

EASY

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

If the normal to the ellipse

{3 x}^{2} + 4 y^{2} = 12

at a point

P

on it is parallel to the line,

2 x + y = 4

and the tangent to the ellipse at

P

passes through

Q (4,4)

then

P Q

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

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,

If a number of ellipses whose major axis is x-axis and the minor axis is y-axis be described having the same length of the major axis as 2 but a variable minor axis, then the tangents at the ends of their latus rectum pass through fixed points whose distance from the centre is equal to

Important Questions on Ellipse

Learn and Prepare for any exam you want !

If a number of ellipses whose major axis is $x$ -axis and the minor axis is $y$ -axis be described having the same length of the major axis as $2$ but a variable minor axis, then the tangents at the ends of their latus rectum pass through fixed points whose distance from the centre is equal to