Find the equations of the tangent and normal to the parabola y 2 = 4ax at the point (at 2 , 2at).

Question

Philoid · Accepted Answer

The equation of a parabola is y² = 4ax, then,

On differentiating it with respect to x, we get

2y

Then, the slope of the tangent at (at², 2at) is

Then, the equation of the tangent at (at², 2at) is given by,

y – 2at =

⇒ ty -2at² = x –at²

⇒ ty = x + at2

Now, Then, slope of normal at (at², 2at)

=

Then, the equation of the normal at (at², 2at) is given by:

y – 2at = -t(x-at²)

⇒ y – 2at = -tx + at³

⇒ y = -tx + 2at + at³

Therefore, the equation of the normal at (at², 2at) is y = -tx + 2at + at³.