Find common normals of the curves $y=1/x^2$ and $x^2 + y^2 - y = 0$

Question

Find common normals of the curves $y=1/x^2$ and $x^2 + y^2 - y = 0$

How we can find the point of contact of the curves because when I solve the both curve the I got irrational roots.

Answer 1

Hint:

The second curve is the circle centred at $(0,\frac12)$, so its normals are its diameters. Hence you have to find the normals to the first curve which pass through the centre of the circle.

Answer 2

Hint.-The tangent at the point $\left(x_0,\dfrac{1}{x_0^2}\right)$ of the curve $y=\dfrac{1}{x^2}$ has the equation $$\frac{y-y_0}{x-x_0}=-\frac{2}{x_0^3}$$ then the corresponding normal has equation $$\frac{y-y_0}{x-x_0}=\frac{x_0^3}{2}$$This line must pass by the center $(0,\frac12)$ of the circle so you get the equation giving the possible values of $x_0$.