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  4. OA and OB are two mutually perpendicular chords of y2 = 4ax, 'O' being the origin. Line AB will always pass through the point

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Q. $OA$ and $OB$ are two mutually perpendicular chords of $y^2 = 4ax$, '$O$' being the origin. Line $AB$ will always pass through the point

Conic Sections

Solution:

Let $A \equiv (at_1 ^2 , 2at_1 ), B \equiv (at_2^ 2 , 2at_2 ) $
Thus $t_1 t_2 = - 4 $
Equation of line $AB $ is $2x - (t_1 + t_2 )y + 2at_1 t_2 = 0$
$2x - (t_1 + t_2 )y - 8a = 0 $
$2(x - 4a) +\lambda y = 0 $
Hence line $AB$ passes through a fixed point $(4a, 0)$