Find the equation of Hyperbola??

Question

Find the equation of the hyperbola that have Foci $F_1(5,0)$ and $F_2(-5,0)$ and the constant difference between the focal radii equal to $8$.

How do I approach this?

Answer 1

Your Focci have coordinates $F_1=(5, 0)$ and $F_1=(-5, 0)$ therefore: $$2c = 5-(-5) = 10 \Rightarrow c = 5$$

The constant difference between the focal radii equal to 8 means that: $$2a = 8 \Rightarrow a = 4$$

You can define that: (look at http://mathworld.wolfram.com/Hyperbola.html) $$b^2 = c^2-a^2$$ $$b = \sqrt{5^2-4^2} = 3$$

The equation of the hyperbole is: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ and therefore the equation for the given hyperbole is: $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$

Answer 2

You can also try the definition

$$|d(P,F_1)-d(P,F_2)|=2a=8$$ where $P=(x,y)$, $F_1=(5,0)$ and $F_2(-5,0)$

Answer 3

\begin{align*} 2a &=8 \\ a &= 4 \\ \sqrt{a^2+b^2} &= 5 \\ b &= 3 \\ \frac{x^2}{16}-\frac{y^2}{9} &= 1 \end{align*}