# Thread: Hyperbola

1. ## Hyperbola

Hyperbola is at the center of coordinate system. Its foci (?) are on abscissa. There are two points on the hyperbola; these are $\displaystyle T_1(\frac{3\sqrt5}{2},2)$ and $\displaystyle T_2(4,\frac{4\sqrt7}{9})$. What is hyperbola's equation?
(I'm having trouble with solving two equations that follow.)

2. Originally Posted by courteous
Hyperbola is at the center of coordinate system. Its foci (?) are on abscissa. There are two points on the hyperbola; these are $\displaystyle T_1(\frac{3\sqrt5}{2},2)$ and $\displaystyle T_2(4,\frac{4\sqrt7}{9})$. What is hyperbola's equation?
(I'm having trouble with solving two equations that follow.)
Since its foci are on the abscissa [x-axis] and the hyperbola is centered at the origin, we know that the hyperbola has the form $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $\displaystyle a>b$

Now plug the two points into this equation, and you will generate two new equations:

At $\displaystyle \left(4,\frac{4\sqrt{7}}{9}\right)$, we get $\displaystyle \frac{16}{a^2}-\frac{112}{81b^2}=1$

At $\displaystyle \left(\frac{3\sqrt{5}}{2},2\right)$, we get $\displaystyle \frac{45}{4a^2}-\frac{4}{b^2}=1$

After solving the system of equations, I got $\displaystyle a=\sqrt{\frac{981}{53}}$

But I'm getting $\displaystyle b^2=-\frac{1744}{171}\implies b\text{ is complex.}$

Are you sure that you wrote down the coordinates correctly?

--Chris

3. Originally Posted by Chris L T521
Are you sure that you wrote down the coordinates correctly?
Sorry!!! No!!! Sorry, Chris! The points are $\displaystyle T_1(\frac{3\sqrt5}{2},2)$ and $\displaystyle T_2(4,\frac{4\sqrt7}{3})$. The $\displaystyle T_2$ y-coordinate's denominator is 3 (not 9).
I've done it so many times that I've automatically restarted with partially already calculated number.