How can I determine wheter or not two curves are tangents?

Question

Let $a_0,b_0 \in \mathbb{R}, a_0^2 + b_0^2 = 1$. How can I show that the curves $$\frac{x^2}{a_0^2} + \frac{y^2}{b_0^2} = 1$$ and $$x+y = 1$$ are tangents to eachother? I tried to find a point such that its gradient vector is equal for both curves (ie,considering $f(x,y) = \frac{x^2}{a_0^2} + \frac{y_2}{b_0^2}$ and $g(x,y) = x+y$). Am I wrong?

Answer 1

Solving simultaneously leads to the quadratic equation $$x^2(a^2+b^2)-2a^2x+a^2-a^2b^2=0$$

The discriminant can be simplified as $$4a^2b^2(a^2+b^2-1)$$

Tangency means double roots, and this is zero since $a^2+b^2=1$

Hence the result.