Locus of the variable point P (x, y) whose distance from a given point A and the given line D =0 is equal then locus is called a “parabola”. Given point A is called focus of the parabola and given line D = 0 is called directrix. A line perpendicular to the directrix and passing through focus is called the ‘axis’ of the parabola. Let A, B, C be three points on the axis of a standard parabola y2 = 4ax whose axis is along the x-axis and A is the focus. A variable chord X Y passing through C subtends 90° at the vertex of the parabola and for another variable chord P Q passing through B, 
is constant.
(i) C point is –
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Ans.
(i)
Sol. Let x & y points are (at 1 2 , 2at 1 ) and
(at 2 2 , 2at 2 ).
t 1 t 2 = – 4 … (1) ( ∠ x o y = 90º)
xy = (t 1 + t 2 ) y = 2x + 2a t 1 t 2
which passes through the point ( 4 a, 0)
(ii)
Sol.. Let mid point is (h, k)
h = 
…(2)
k = 
… (3)
By using (1), (2) & (3) eliminate t 1 & t 2 .
(iii)
Sol. Let B : (x 1 , 0) point and line through B makes angle θ with +ve x-axis.
r 1 and r 2 are the roots of
r 2 sin 2 θ – 4 (x 1 + r cos θ ) = 0
= 
which is independent of ' θ ' for x 1 = 2a.
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