Question

The condition that the parabolas $y^2=4ax$ and $y^2=4c(x-b)$ 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)

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

For parabola $y^2=4ax$,

Equation of normal: $y=mx-2am-a{m}^3$ ...(1)

And for parabola $y^2=4c(x-b)$,

Equation of normal: $y=m(x-b)-2cm-c{m}^3$ ...(2)

Since the two parabolas have a common normal, therefore (1) and (2) must be identical.

After comparing the coefficients, we get

$m=\pm \sqrt{\frac{2(a-c)-b}{(c-a)}}$

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

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