Q.99. Tangents to a parabola from an external point, multipe-choice question.

Question 99.
If two tangents drawn from the point (m,  n) to the parabola y^2 = 4x be such that the slope of one tangent is twice that of the other, then select the right option.
a) n = 2/9 * m^2          b) m = 2/9* n^2          c) 2m = 9n^2          d) none of these

Answer 99.
If a = slope of one tangent, 2a is that of the other
If a and 2a are the roots of a quadratic in t, its equation is
t^2 - 3at + 2a^2 = 0 ... (1)

 The equation of a tangent to the parabola y^2 = 4x having slope t is
y = tx + 1/t
If it passes through (m, n),
n = mt + 1/t
=> mt^2 - nt + 1 = 0 ... (2)

 Eqns. (1) and (2) must be the same
=> m = n/3a = 1/2a^2
=> a = 3m/n and a^2 = 2m
=> 2m = (3m/n)^2
=> 2 = 9m/n^2
=> m = (2/9)n^2
=> Answer is b) m = 2/9*n^2.

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