# Finding equation of hyperbola from known coordinates of focii

• Jan 27th 2009, 09:31 AM
Rinor Berisha
Finding equation of hyperbola from known coordinates of focii
I've been having problems with one particular one. here it is
"Find the equation of the hyperbola if its focuses are $\displaystyle F_1=(10,-2)F_2=(16,2)$ and $\displaystyle 2a=24$.

Any help is greatly appreciated

Regards,
Rinor
• Jan 28th 2009, 06:32 AM
earboth
Quote:

Originally Posted by Rinor Berisha
I've been having problems with one particular one. here it is
"Find the equation of the hyperbola if its focuses are $\displaystyle F_1=(10,-2)F_2=(16,2)$ and $\displaystyle 2a=24$.

Any help is greatly appreciated

Regards,
Rinor

The only way I found is: Use the definition of the hyperbola.

Let P denote any arbitrary point on the hyperbola. Then the equation must be true:

$\displaystyle |\overline{PF_1}| - |\overline{PF_2}| = 2a$

But I hope for you that the original question reads:

"Find the equation of the hyperbola if its focuses are $\displaystyle F_1={\bold{(-10,2)}},~F_2=(16,2)$ and $\displaystyle 2a=24$.

If so:
1. The center is at C(3, 2)
2. e = 13
3. From $\displaystyle e=\sqrt{a^2+b^2}$ follows b = 5
4. The equation of the hyperbola is: ....... $\displaystyle \dfrac{(x-3)^2}{144}-\dfrac{(y-2)^2}{25}=1$