Position of point with respect to hyperbola

Question

I know we can find position of point with respect to hyperbola using put the coordinate of point to formula of hyperbola . If the answer is negative then it's outside and if positive then it's inside . Why this is true ?

Answer 1

Let hyperbola be $S\equiv\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Let $P(x,y_1)$ be outside the hyperbola and $Q(x,y_2)$ be on the hyperbola. Then, $y_1^2>y_2^2$ (verify by drawing an image!)

Now, $$y_1^2>y_2^2$$$$y_1^2>(\frac{x^2}{a^2}-1)b^2$$$$\frac{y_1^2}{b^2}>\frac{x^2}{a^2}-1$$$$0>\frac{x^2}{a^2}-\frac{y_1^2}{b^2}-1$$

This shows that $S_P<0$ for point P outside the hyperbola. Similarly, one can prove $S_P>0$ for point P inside the hyperbola.