Finding the eq of a normal to the parabola

Find the eq of the normal at a pt on the parabola $\displaystyle y^2 = 2x$ whose ordinate is 4

Solution

Eq of tangent to parabola at $\displaystyle (x_1,y_1) $ is $\displaystyle \Rightarrow yy_1 = 2a(x+x_1)$

Eq of parabola is $\displaystyle y^2 = 2x \therefore a = \frac{1}{2}$

\therefore the eq of a tangent to the parabola at $\displaystyle (x_1,4)$ is $\displaystyle \Rightarrow 4y = (x+x_1) \Rightarrow y = \frac{1}{4}(x+x_1)$

since the normal i perpendicular to the tangent then its slope would be -4

$\displaystyle \therefore$ eq of the normal to the parabola at a pt $\displaystyle (x_1, 4) $ is

$\displaystyle y = -4 (x+x_1)$

**Am i correct !!!!!**