Question

The condition that the parabolas y² = 4ax and y² = 4c(x – b) have a common normal other than x-axis (a, b, c being distinct positive real numbers) is-

• A. $\frac{b}{a - c} < 2$
• B. $\frac{b}{a - c} > 2$
• C. $\frac{b}{a - c} < 1$
• D. $\frac{b}{a - c} > 1$

Answer 1

Answer: (B)

$\frac{b}{a - c} > 2$

Answer 2

y² = 4ax eqⁿ of normal

y = mx – 2am – am³

y² = 4c(x –b) eqⁿ of normal

y = m(x – b) – 2cm – cm³

It two parabola have common normal then both of eqⁿ of normal should be identical after comparing the coefficients

m = ± $\sqrt{\frac{2(a - c) - b}{(c - a)}}$ which is real if

– 2 – $\frac{b}{c - a}$> 0 ̃ $\frac{b}{a - c}$ > 2