JEE MAIN - Mathematics (2019 - 12th April Evening Slot - No. 15)
An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the following points?
$$\left( {2,\sqrt 2 } \right)$$
$$\left( {2,2\sqrt 2 } \right)$$
$$\left( {\sqrt 2 ,2} \right)$$
$$\left( {1,2\sqrt 2 } \right)$$
Explanation
Let $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1(a < b)$$ is the equation of ellipse, focii $$(0, \pm 2)$$
Given 2a = 4 $$ \Rightarrow $$ a = 2
e2 = 1 - $${{{a^2}} \over {{b^2}}}$$ $$ \Rightarrow $$ b2e2 = b2 - a2
4 = b2 - 4
b2 = 8
$$ \because $$ equation of ellipse is $${{{x^2}} \over 4} + {{{y^2}} \over 8} = 1$$
then it passes through $$\left( {\sqrt 2 ,2} \right)$$
Given 2a = 4 $$ \Rightarrow $$ a = 2
e2 = 1 - $${{{a^2}} \over {{b^2}}}$$ $$ \Rightarrow $$ b2e2 = b2 - a2
4 = b2 - 4
b2 = 8
$$ \because $$ equation of ellipse is $${{{x^2}} \over 4} + {{{y^2}} \over 8} = 1$$
then it passes through $$\left( {\sqrt 2 ,2} \right)$$
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