QJEEM21Macd56: JEE Main 2021 Mathematics - Class 11 JEEM21Macd56 Solution: Let an ellipse , , passes through and has eccentricity . If a…

Question

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Let an ellipse

E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1

,

{a^2} &gt; {b^2}

, passes through

\left( {\sqrt {{3 \over 2}} ,1} \right)

and has eccentricity

{1 \over {\sqrt 3 }}

. If a circle, centered at focus F(

\alpha

, 0),

\alpha

> 0, of E and radius

{2 \over {\sqrt 3 }}

, intersects E at two points P and Q, then PQ² is equal to :

Accepted Answer

{3 \over {2{a^2}}} + {1 \over {{b^2}}} = 1

and

1 - {{{b^2}} \over {{a^2}}} = {1 \over 3}

\Rightarrow {a^2} = 3{b^2} = 3

\Rightarrow {{{x^2}} \over 3} + {{{y^2}} \over 2} = 1

...... (i)

Its focus is (1, 0)

Now, equation of circle is

{(x - 1)^2} + {y^2} = {4 \over 3}

..... (ii)

Solving (i) and (ii) we get

y = \pm {2 \over {\sqrt 3 }},x = 1

\Rightarrow P{Q^2} = {\left( {{4 \over {\sqrt 3 }}} \right)^2} = {{16} \over 3}

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