If the eccentricity of the hyperbola $x^2-y^2 \operatorname{coces}^2 \alpha=25$ is $\sqrt{5}$ times the eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \alpha+y^2=5$, then $\alpha$ is equal to :

$\sin ^{-1} \sqrt{\frac{3}{4}}$

$\tan ^{-1} \sqrt{\frac{2}{5}}$

$\sin ^{-1} \sqrt{\frac{2}{5}}$

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JEE MATHEMATICS MCQ: ELLIPSE

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Application +4 | -1

If the eccentricity of the hyperbola

x^2-y^2 \operatorname{coces}^2 \alpha=25

\sqrt{5}

times the eccentricity of the ellipse

x^2 \operatorname{cosec}^2 \alpha+y^2=5

, then

\alpha

is equal to :

View answer and explanation

Correct option: A

\tan ^{-1} \sqrt{2}

Option A (

\tan^{-1} \sqrt{2}

) is correct because by equating the eccentricities of the hyperbola and the ellipse as given in the question, we can solve for

\alpha

and find that

\alpha = \tan^{-1} \sqrt{2}

More Questions from ELLIPSE

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Let

\frac{x^2}{144}+\frac{y^2}{81}=1

be the equation of an ellipse. If

m

be the minimum value of the segment of tangent to the ellipse intercepted by the coordinate axes and

\mathrm{M}

be the maximum distance of the centre of the ellipse from the normal, then the value of

\mathrm{mM}

is equal to Let

P

be a variable point on the ellipse

\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1

with foci at

S

and

S'.

A

be the area of triangle

PSS',

then the maximum value of

A

is If the co-ordinates of two points

A

and

B

are

(\sqrt{7}, 0)

and

(-\sqrt{7}, 0)

respectively and

P

is any point on the conic,

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

, then

\mathrm{PA}+\mathrm{PB}

is equal to :
The slope of a common tangent to the ellipse

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

and a concentric circle of radius

r

, is A tangent to the ellipse

\frac{x^2}{25}+\frac{y^2}{16}=1

at any point

P

meet the line

x=0

at a point

Q

. Let

R

be the image of

Q

in the line

y=x

, then circle whose extremities of a diameter are

Q

and

R

passes through a fixed point. The fixed point is Any ordinate

N P

of an ellipse meets the auxiliary circle at

Q

. Then, the locus of the intersection of the normal at

P

and

O

is the circle