# How do you write a standard form equation for the hyperbola with foci are (-6,0) and (6,0) and the difference of the focal radii is 10?

Apr 23, 2016

${x}^{2} / {5}^{2} - {y}^{2} / \left({\left(\sqrt{11}\right)}^{2}\right) = 1$ or $11 {x}^{2} - 25 {y}^{2} - 275 = 0$

#### Explanation:

As both the foci are on the x-axis, x-axis is the major axis of the hyperbola.

The difference between the focal radii = length of the major axis 2a = 10. So, a = 5.

The distance between the foci s(6, 0) and S'#(-6, 0) = 12 = major axis length 2 a x eccentricity = 2 a e = 10 e =12. So, e = 6/5.

The semi-transverse axis $b = a \sqrt{{e}^{2} - 1} = \sqrt{11}$

The standard form of the equation is ${x}^{2} / {a}^{2} - {y}^{2} / {b}^{2} = 1$
here, it is
${x}^{2} / {5}^{2} - {y}^{2} / \left({\left(\sqrt{11}\right)}^{2}\right) = 1$ or $11 {x}^{2} - 25 {y}^{2} - 275 = 0$.