# Hyperbola

A Hyperbola is the set of all points P(x, y) in the plane such thatAgain F1 and F2 are focus points. This time the difference of these distances remain a constant at 2a. The explanation is similar to that of the ellipse. Since the ellipse is the sum of the distances and the hyper-bola is the difference of the distances, the equations are very similar. They differ only in the sign and the longest side for a hyper-bola is c. (Remember for the ellipse it was a)

The equation of the above hyper-bola is

The hyper-bola opens left and right. Notice it comes in two parts. Different than an ellipse which is a closed figure. Hyper-bolas can also open up and down. So the equation is opposite to that of above hyper-bola.

If the center of the hyper-bola is other than origin. Suppose the center is (h,k)

If the hyper-bola opens right/left the translation is :

If the hyper-bola opens up/down the translation is:

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**Practice**

1) Find the coordinates of the vertices, the foci, the eccentricity and the equations of the directrices of the hyper-bola 4x

^{2}- 25 y

^{2}= 100. (Ans)

2) Find the equation of the hyper-bola with vertices at (~+mn~ 5,0) and foci

at (~+mn~ 7,0). (Ans)

3) Find the equation of the hyper-bola with vertices at (~+mn~ 6,0) and one of the directrix is x = 4. (Ans)

4) Find the coordinates of the vertices, the foci, the eccentricity and the equations of the directrices of the hyper-bola 9x

^{2}- 16 y

^{2}= 144. (Ans)

5) Find the coordinates of the vertices, the foci, the eccentricity and the equations of the directrices of the hyper-bola y

^{2}- 16x

^{2}= 16. (Ans)

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