A focal chord of parabola y2 = 4x is inclined at an angle of with the +ive direction of x-axis, then the slope of normal drawn at the ends of focal chord will satisfy the equation
m2 + 2m – 1 = 0
Let A, B be the points (t12, 2t1) and (t22, 2t2), (a = 1) be two pints on the parabola y2 = 4x.
Since AB is a focal chord, t1 t2 = –1.
Also slope of chord y(t1 + t2) – 2x – 2at1t2 = 0 is
Hence t1, t2 are the roots of
m2 – 2m – 1 = 0
Slopes of normal’s at A and B are – t1, – t2 which are roots of
(–m)2 – 2(–m) – 1 = 0
m2 + 2m – 1 = 0
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