Question

Length of normal chord of parabola $y^2=4x$ which makes an inclination of $\frac{\pi }{4}$ with the positive direction of $x-$axis is

• A. $8$ units
• B. $8\sqrt{2}$ units
• C. $4$ units
• D. $4\sqrt{2}$ units

Answer 1

The correct option is B $8\sqrt{2}$ units

Normal of $y^2=4ax$ at $(a{t}^2,2at)$ is
$y=-tx+a{t}^3+2at$
$\Rightarrow$ slope $=-t=\tan \frac{\pi }{4}$
$\Rightarrow t=-1$
Normal at point $P(t=-1)$ meets $y^2=4x$ again at $Q\left(t_0\right)$
$\Rightarrow t_0=-t-\frac{2}{t}=1+2=3$
$\therefore Q=(a{t}_0^2,2a{t}_0)=(9,6)[\because a=1]$
Also, $P=(1,-2)$
$\therefore$ Length of normal chord $=\sqrt{(9-1{)}^2+(6-(-2){)}^2}=8\sqrt{2}$ units