If circumcentre of an equilateral triangle inscribed in x 2 a 2 + y 2 b 2 = 1 , with vertices having eccentric angles α , β , γ respectively, is x 1 , y 1 , then Σ cos α cos β + Σ sin α sin β =

9 x 1 2 2 a 2 + 9 y 1 2 2 b 2 - 3 2

9 x 1 2 a 2 + 9 y 1 2 b 2 + 3 2

If circum centre of an equilateral triangle inscribed in x 2 a 2 + y 2 b 2 = 1 with vertices having eccentric angles α , β , γ respectively is ( x 1 , y 1 ) then ∑ c o s α c o s β + ∑ s i n α s i n β =

9 x 1 2 2 a 2 + 9 y 1 2 2 b 2 - 3 2

9 x 1 2 a 2 + 9 y 1 2 b 2 + 3 2

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The curve with parametric equations $x = 1 + 4$ $\cos θ, y = 2 + 3 \sin θ$ is

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Important Questions on Ellipse

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

P

is the extremity of the latusrectum of ellipse

3 x^{2} + 4 y^{2} = 48

in the first quadrant. The eccentric angle of

P

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

Let

P

be a point on the ellipse

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

and the line through

P

parallel to the

Y

-axis meets the circle

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

Q,

where

P, Q

are on the same side of the

X

-axis. If

R

is a point on

P Q

such that

\frac{P R}{R Q} = \frac{1}{2},

then the locus of

R

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

The locus of the mid-point of the line segment joining the point

(4, 3)

and the points on the ellipse

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

is an ellipse with eccentricity

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

If the point

P

on the curve,

4 x^{2} + 5 y^{2} = 20

is farthest from the point

Q (0, - 4)

, then

P Q^{2}

is equal to

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

If $A (\frac{3}{\sqrt{2}}, \sqrt{2}), B (- \frac{3}{\sqrt{2}}, \sqrt{2}), C (- \frac{3}{\sqrt{2}}, - \sqrt{2})$ and $D (3 cos θ, 2 \sin θ)$ are four points, the value of $θ$ for which the area of quadrilateral $A B C D$ is maximum, $(\frac{3 π}{2} \leq θ \leq 2 π)$ is equal to

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

The curve with parametric equations

x = α + 5 \cos θ, y = β + 4 \sin θ

(where

θ

is parameter) is

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

The parametric representation of a point on the ellipse whose foci are

(- 1,0)

and

(7, 0)

and eccentricity

\frac{1}{2}

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

\tan θ_{1} \cdot \tan θ_{2} = - \frac{a^{2}}{b^{2}},

then the chord joining two points

P (a \cos θ_{1}, b \sin θ_{1})

and

Q (a \cos θ_{2}, b \sin θ_{2})

on the ellipse

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

will subtend a right angle at

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

If the eccentric angles of the extremities of a focal chord of an ellipse

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

are

α

and

β

, then

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

A point on the ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1

at a distance from centre equal to the mean of the lengths of the semi-major axis and semi-minor axis, is -

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

Let

P

be a point on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, 0 < b < a .

Let the line parallel to

y

-axis passing through

P

meets the circle

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

at the point

Q

such that

P

and

Q

are on the same side of

x

-axis. For two positive real numbers

r

and

s

, find the locus of the point

R

P Q

such that

P R : R Q = r : s

, as

P

varies over the ellipse.

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

If circumcentre of an equilateral triangle inscribed in

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

with vertices having eccentric angles

α, β, γ

respectively, is

(x_{1}, y_{1})

, then

Σ \cos α \cos β + Σ \sin α \sin β =

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

Let

P

be a variable point on the ellipse

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

with foci

S_{1}

and

S_{2}

. If

A

be the area of

Δ {P S}_{1} S_{2}

, then the maximum value of

A

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

θ

and

ϕ

are eccentric angle of the ends of a pair of conjugate diameters of the ellipse

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

, then

θ - ϕ

is equal to

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

If circum centre of an equilateral triangle inscribed in

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

with vertices having eccentric angles

α, β, γ

respectively is

(x_{1}, y_{1})

then

\sum c o s α c o s β + \sum s i n α s i n β =

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

In an ellipse find the locus of point of interaction of the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact.

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

If C is the centre and

A, B

are two points on the conic

4 x^{2} + 9 y^{2} - 8 x - 36 y + 4 = 0

such that

A \hat{C} B = \frac{π}{2}

, then

{C A}^{- 2} + {C B}^{- 2}

is equal to-

MEDIUM

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

The line passing through the extremity

A

of the major axis and extremity

B

of the minor axis of the ellipse

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

meets its auxiliary circle at the point

M .

Then the area of the triangle with vertices at

A, M

and the origin

O

EASY

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

The equation of the auxiliary circle of

\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1

HARD

Mathematics>Coordinate Geometry>Ellipse>Parametric Equations of an Ellipse

A (1, 4)

and

B (3, 0)

, then the maximum area of triangle

A B C

inscribed in the ellipse

2 x^{2} + y^{2} = 18

The curve with parametric equations x=1+4 cosθ,y=2+3sinθ is

Important Questions on Ellipse

Learn and Prepare for any exam you want !

The curve with parametric equations $x = 1 + 4$ $\cos θ, y = 2 + 3 \sin θ$ is