Question

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

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

Answer 1

Answer: (B)

$8\sqrt{2}$

Given the equation of parabola is
$y^2=4x$

Here, $a=1$

Let $P(t_1^2,2t_1)$ and $Q(t_2^2,2t_2)$ be the endpoints of normal chord of the parabola.

Equation of normal at point $(t_1^2,2t_1)$ is $y=-t_{1}x+2t_1+{t_1}^3$

The slope of the normal chord is $1$.

Hence, $-t=1\Rightarrow t=-1$

$\Rightarrow$ Coordinates of $P$ are $(1,-2)$.

Hence, parameter at $Q$ is $t_2=-t_1-\frac{2}{t_1}=1+2=3$

Therefore, coordinates at $Q$ are $(9,6)$

Length of $PQ=\sqrt{8^2+8^2}=8\sqrt{2}$