Question

The normal y = mx – 2am – am2 to the parabola y2 = 4ax subtends a right angle at the origin, then-

• A. m = 1
• B. m = $\sqrt{2}$
• C. m = 2
• D. m = $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (B)

m = $\sqrt{2}$

Answer 2

Standard equation of the normal at P(at2, 2at) is y + xt
= 2at + at3.

Hence t = – m. This meets the parabola again at Q whose parameter is – t – $\frac{2}{t}$ or m +$\frac{2}{m}$

\ coordinates of Q are a $\left( m + \frac{2}{m} \right)^{2}$, 2a $\left( m + \frac{2}{m} \right)$

Since ĐPOQ = 90⁰ ̃$\left( \frac{2}{m + \frac{2}{m}} \right)\left( \frac{2}{–m} \right)$ = –1

̃ m2 + 2 = 4

̃ m = ±$\sqrt{2}$

m = $\sqrt{2}$ is one of the choices.