Point of intersection between two curves:
$ax^2+by^2=1$ and ${a}'x^2+{b}'y^2=1$. Let $P(h,k)$ be point of intersection $\Rightarrow$
$ah^2+bk^2=1$ and ${a}'h^2+{b}'k^2=1$
Now, $$\frac{h^2}{-b+{b}'}=\frac{k^2}{-a+{a}'}=\frac{1}{a{b}'-{a}'b}$$
How was this arrived at ? What is the general form for this?
multiply one equation (1) by $a'$ and equation (2) by $a$, subtract and rearrange the terms. Repeat with $b$ and $b'$.