Question

The locus of middle point of the portion of the normal to $y^2=4ax$ intercepted between curve and axis of parabola is

• A. $y^2=a(x-a)$
• B. $y^2=2a(x-a)$
• C. $y^2=(x-2a)$
• D. $y^2=a^2(x-2a)$

Answer 1

The correct option is A $y^2=a(x-a)$

Given parabola is $y^2=4ax$
Equation of normal at $P(a{t}^2,2at)$ is
$tx+y=2at+a{t}^3$
Point of intersection of normal with axis of parabola is
$\Rightarrow Q=(2a+a{t}^2,0)$


Let the midpoint be $R(h,k)$, so
$h=\frac{2a+a{t}^2+a{t}^2}{2},k=\frac{2at}{2}\Rightarrow h=a+a{t}^2,k=at\Rightarrow h=a+\frac{a{k}^2}{a^2}\Rightarrow ah=a^2+k^2$

Hence, the required locus is
$y^2=a(x-a)$