Question

Find the equation of the tangent and the normal to the given curve at the indicated point: y² = 4ax at (at², 2at)

Answer 1

Consider y² = 4ax as the equation of curve

By differentiating both sides w.r.t. x

2y dy/dx = 4a

We can write it as

dy/dx = 2a/y

By substituting the values

(dy/dx)_(at²_{, 2at)} = 2a/2at = 1/t

Here the equation of tangent at point (at², 2at)

y – 2at = 1/t(x – at²)

On further calculation

yt – 2at² = x – at²

So we get

x – ty + at² = 0

Here the equation of normal at point (at², 2at)

y – 2at = – t(x – at²)

By further calculation

y – 2at = – tx + at³

We can write it as

y + tx = at³ + 2at

So we get

tx + y = at³ + 2at