# Lec 13 Fig 1: Foci of a hyperbola

A hyperbola is a locus of points $$A(x,y)$$ whose difference of its distances from two fixed points $$F_1$$ and $$F_2$$ (called the foci, located at $$(c,0)$$ and $$(-c,0)$$ in the diagram) is constant ($$K$$ in the diagram). The following diagram shows all possible locations of $$A$$ in that locus. Try to adjust the values $$c$$ and $$K$$ and move the point $$A$$ to see the locus.

If we set the foci at $$F_1(c,0)$$ and $$F_2(-c,0)$$, then the locus of $$A(x,y)$$ can be defined by $|AF_2| - |AF_1| = K$ To find what $$K$$ means, we look at the special point $$A(a,0)$$. In this case, $$|AF_1| = a-c$$ and $$|AF_2| = a+c$$, which means $$K = 2a$$, the distance from the centre (midpoint of the foci) to the point on the focus axis. For general $$A(x,y)$$, $|AF_2| - |AF_1| = 2a \quad \Rightarrow \quad \sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a.$ After some serious simplication, we get $\dfrac{x^2}{a^2} - \dfrac{y^2}{c^2-a^2} = 1.$ Let $b^2 = c^2 - a^2,$ then we have the standrd form of the equation of a hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1.$