If ℓ 1 be the equation of the common tangent in 1 st quadrant to the circle x 2 + y 2 = 16 and ellipse x 2 25 + y 2 4 = 1 and λ 1 be the length of the intercept of the common tangent between the coordinate axes then

Equation of ℓ 1 is 2 x + 3 y = 4 7 .

Equation of ℓ 1 is x + 3 y = 4 7 .

In a triangle A B C with fixed base B C , the vertex A moves such that cos B + cos C = 4 sin 2 A 2 . If a , b and c denote the lengths of the sides of the triangle opposite to the angles A , B and C , respectively, then

Iocus of points A is an ellipse

locus of point A is a pair of straight lines

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Let $F_{1}$ and $F_{2}$ be two foci of the ellipse and $P T$ and $P N$ be the tangent and the normal respectively to the ellipse at point $P .$ Then,

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Important Questions on Ellipse

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-2>Q 22

ℓ_{1}

be the equation of the common tangent in

1^{st}

quadrant to the circle

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

and ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1

and

λ_{1}

be the length of the intercept of the common tangent between the coordinate axes then

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-2>Q 23

Let

E_{1}

and

E_{2}

be two ellipses

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

and

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

(where

a

is a parameter). Then the locus of the points of intersection of the ellipses

E_{1}

and

E_{2}

is a set of curves

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-2>Q 24

(5, 12)

and

(24, 7)

are the foci of a conic passing through origin, then eccentricity of the conic is

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 1

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 the axillary circle at the point

M

, then the area of the triangle with vertices at

A

M

and the origin

O

N

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 2

In a triangle

A B C

with fixed base

B C,

the vertex

A

moves such that

\cos B + \cos C = 4 \sin^{2} \frac{A}{2} .

a, b

and

c

denote the lengths of the sides of the triangle opposite to the angles

A, B

and

C

, respectively, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 3

The normal at a point

P

on the ellipse

x^{2} + 4 y^{2} = 16

meets the

x

-axis at

Q

. If

M

is the mid-point of the line segment

P and Q

, then the locus of

M

intersects the latus rectum of the given ellipse at the points.

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 5

The ellipse

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

is inscribed in a rectangle

R

whose sides are parallel to the coordinate axes. Another ellipse

E_{2}

passing through the point

(0, 4)

circumscribes the rectangle

R

. The eccentricity of the ellipse

E_{2}

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Alpha Question Bank for Engineering: Mathematics>Chapter 18 - Ellipse>Exercise-3>Q 6

Suppose that the foci of the ellipse

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

are

(f_{1}, 0) a n d (f_{2}, 0)

where

f_{1} > 0 a n d f_{2} < 0 .

Let

P_{1} a n d P_{2}

be two parabolas with a common vertex at (0, 0) and with foci at

(f_{1}, 0)

and

(2 f_{2}, 0),

respectively. Let

T_{1}

be a tangent to

P_{1}

which passes through

(2 f_{2}, 0)

and

T_{2}

be a tangent to

P_{2}

which passes through

(f_{1}, 0)

. If

m_{1}

is the slope of

T_{1}

and

m_{2}

is the slope of

T_{2},

then the value of

(\frac{1}{m_{1}^{2}} + m_{2}^{2})

Let F1 and F2 be two foci of the ellipse and PT and PN be the tangent and the normal respectively to the ellipse at point P. Then,

Important Questions on Ellipse

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Let $F_{1}$ and $F_{2}$ be two foci of the ellipse and $P T$ and $P N$ be the tangent and the normal respectively to the ellipse at point $P .$ Then,