Let the hyperbola H:a2x2−b2y2=1 pass through the point (22,−22). A parabola is drawn whose focus is the same as the focus of H with positive abscissa, and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?
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Official solution
\[H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\]
Focus of parabola : \[(ae,\,0)\]
Directrix : x=−ae.
Equation of parabola \[ \equiv {y^2} = 4aex\]
Length of latus rectum of parabola =4ae
Length of latus rectum of hyperbola \[ = {{2.{b^2}} \over a}\]
as given, \[4ae = {{2{b^2}} \over a}\,.\,e\]
\[2 = {{{b^2}} \over {{a^2}}}\] ...... (i)
\[\because\] H passes through \[\left( {2\sqrt 2 , - 2\sqrt 2 } \right) \Rightarrow {8 \over {{a^2}}} - {8 \over {{b^2}}} = 1\] ........ (ii)
From (i) and (ii) a2=4 and \[{b^2} = 8 \Rightarrow e = \sqrt 3 \]
\[\Rightarrow\] Equation of parabola is \[{y^2} = 8\sqrt 3 x\].