Question

What will be the equations of normal to the parabola $y^2=4x$ at the ends of latus rectum?

Answer 1

Firstly, the coordinates of end points of latus rectum for the parabola $y^2=4x$ are $A(1,2)\&B(1,-2).$
Secondly, we have to find the slope of normal at end points.
So, the slope of normal at any point $(x_1,y_1)$ on the parabola $y^2=4ax$ be $=-\frac{y_1}{2a}$
$\therefore$ Slope at point A is $=-\frac{2}{2}=-1$
And, Slope at point B is $=-\frac{(-2)}{2}=1$
Thus, we got both the points and both the slope of normal at that points.
Now, applying the point form to each point we get,
At A$:y-2=(-1)(x-1)$
$\Rightarrow x+y=3$
At B$:y+2=1(x-1)$
$\Rightarrow x-y=3$
Hnece, the equation of normals at the end points of latus rectum are $x+y=3$ and $x-y=3.$