First slide

Parabola

Question

$T h e s t r a i g h t l i n e j o i n i n g a n y p o i n t P o n t h e p a r a b o l a y^{2} = 4 a x t o t h e v e r t e x a n d p e r p e n d i c u l a r f r o m t h e f o c u s t o t h e \tan g e n t a t P, i n t e r s e c t a t R, t h e n t h e c q u a t i o n o f t h e l o c u s o f R i s -$

Moderate

A

$x^{2} + 2 y^{2} - a x - 0$
B

$2 x^{2} + y^{2} - 2 a x - 0$
C

$2 x^{2} + 2 y^{2} - a y = 0$
D

$2 x^{2} + y^{2} - 2 a y = 0$

Solution

$Equation of tangent a t P (a t^{2}, 2 a t) is ty = x + {at}^{2} - - - - - - - (1)$
$Line perpendicular to (1) through (a, 0)$
$t x + y = t a - - - - - - - - (2)$
$Equation of OP: y - \frac{2}{t} x = 0 - - - - - - - (3)$
$From equations (2) and (3) eliminating t we get locus$
$2 x^{2} + y^{2} - 2 a x = 0$

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