The curve below is given by the equations

where

1. Prove that the tangents to the curve at the points $A$ and $B$ intersect at the point $C$

2. Show that a Cartesian equation for the curve is

   $$y^2(x^2+1)=(x+1{)}^2$$

   and hence, using implicit differentiation, show that

   $$\frac{dy}{dx}=\frac{x(1-y^2)+1}{y(1+x^2)}$$

For (a), try not to make any assumptions. All we know is that $A,B$ have $x=0$ and $C$ has $y=0$.

Carefully find the equations of the tangents at $A$ and $B$ and then find their point of intersection. If this is equal to the point $C$, then you're finished.

For (b), try substituting $x$ and $y$ into the left hand side of

and try to show that it becomes equal to the right hand side.

When doing the implicit differentiation, you will need the chain rule and the product rule.