HARD

Earn 100

Prove that the minimum length of the intercept made by the axes on the tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is equal to $a + b$ .

Important Questions on Ellipse

MEDIUM

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

MEDIUM

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

If the line

y = 2 x + c

touches the curve

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

then

c^{2} =

MEDIUM

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

HARD

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

Let

L

be a common tangent line to the curves

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

and

{(2 x)}^{2} + {(2 y)}^{2} = 31

. Then the square of the slope of the line

L

is ______.

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to a 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:

MEDIUM

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

For

0 < θ < \frac{π}{2}

, four tangents are drawn at the four points

(\pm 3 \cos θ, \pm 2 \sin θ)

to the ellipse

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

A (θ)

denotes the area of the quadrilateral formed by these four tangents, the minimum value of

A (θ)

MEDIUM

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

EASY

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

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

HARD

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

MEDIUM

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

For which of the following curves, the line

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

is the tangent at the point

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

MEDIUM

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

The values of

λ

for which the line

y = x + λ

touches the ellipse

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

are

EASY

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to a 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 a 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:

HARD

Mathematics>Coordinate Geometry>Ellipse>Tangent and Normal to a 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 a 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,

Prove that the minimum length of the intercept made by the axes on the tangent to the ellipse x2a2+y2b2=1 is equal to a+b.

Important Questions on Ellipse

Learn and Prepare for any exam you want !

Prove that the minimum length of the intercept made by the axes on the tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is equal to $a + b$ .