Let an ellipse [E:x^2 a^2 + y^2 b^2 = 1], a^2 b^2, passes through [( 3 2 ,1 )] and has e…

Question

Let an ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, ${a^2} > {b^2}$, 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

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

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

Solutions · 1