MEDIUM

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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:

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Important Questions on Conic sections

MEDIUM

Mathematics>Coordinate Geometry>Conic sections>Tangent to a Ellipse

β

is one of the angles between the normals to the ellipse

x^{2} + 3 y^{2} = 9

at the points

(3 \cos θ, \sqrt{3} \sin θ)

and

(- 3 \sin θ, \sqrt{3} \cos θ); θ \in (0, \frac{π}{2})

; then

\frac{2 \cot β}{\sin 2 θ}

is equal to :

HARD

Mathematics>Coordinate Geometry>Conic sections>Tangent 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>Conic sections>Tangent 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>Conic sections>Tangent to a 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>Conic sections>Tangent 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>Conic sections>Tangent to a 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>Conic sections>Tangent to a 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

MEDIUM

Mathematics>Coordinate Geometry>Conic sections>Tangent to a 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

HARD

Mathematics>Coordinate Geometry>Conic sections>Tangent 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

HARD

Mathematics>Coordinate Geometry>Conic sections>Tangent 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?

MEDIUM

Mathematics>Coordinate Geometry>Conic sections>Tangent 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>Conic sections>Tangent 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,

HARD

Mathematics>Coordinate Geometry>Conic sections>Tangent 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>Conic sections>Tangent to a 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>Conic sections>Tangent 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

HARD

Mathematics>Coordinate Geometry>Conic sections>Tangent 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>Conic sections>Tangent 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

EASY

Mathematics>Coordinate Geometry>Conic sections>Tangent to a 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:

HARD

Mathematics>Coordinate Geometry>Conic sections>Tangent to a 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>Conic sections>Tangent to a Ellipse

The value of

λ

for which the line

y = x + λ

touches the ellipse

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

is/are

The tangent and normal to the ellipse 3x2+5y2=32 at the point P2, 2 meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is:

Important Questions on Conic sections

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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: