ParabolaHard
Question
If two tangents drawn from a point P to the parabola y2 = 4x be such that the slope of one tangent is double of the other, then P lies on the curve. :-
Options
A.9y = 2x2
B.9x = 2x2
C.2x = 9y2
D.None of these
Solution
Let P(h, k) be the point from which two tangents are drawn to y2 = 4x. Any tangent to the parabola y2 = 4x is
y = mx +
If it passes through P(h, k), then
k = mh +
⇒ m2 h - mk + 1 = 0
Let m1, m2 be the roots of this equation. Then,
m1 + m2 =
and m1m2 = 
⇒ 3m2 =
and 2m22 = 
[∵ m1 = 2m2(given)]
⇒ 2
⇒ 2k2 = 9h
Hence, P(h, k) lies on 2y2 = 9x
y = mx +
If it passes through P(h, k), then
k = mh +
⇒ m2 h - mk + 1 = 0Let m1, m2 be the roots of this equation. Then,
m1 + m2 =
and m1m2 = ⇒ 3m2 =
and 2m22 = [∵ m1 = 2m2(given)]⇒ 2
⇒ 2k2 = 9hHence, P(h, k) lies on 2y2 = 9x
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