Question

Show that the radius of curvature x^(2/3) + y^(2/3) = a^(2/3) at P(x, y) is three times the distance from the origin to the tangent line T.

Answer 1

The given curve is

Differentiating both side w.r.t x

Again differentiating both sides w.r.t x :

Let the point p be , then the curvature at p is

i.e.

i.e.

Thus radius of curvature at P is

…………… (1)

Now, we have

Then the slope of tangent T at is

Therefore, the equation of tangent line T is

Or,

Or,

Now, the distance of origin (0,0) from this tangent line is

Since lies on the curve than

Thus

From (1) and (2), replace , then

This gives

i.e. the radius of curvature is there time the distance from the origin to the tangent line T