Class 11

Math

3D Geometry

Conic Sections

Find the equation of the hyperbola satisfying the give conditions: Vertices $(0,±5)$ foci $(0,±8)$

Here the vertices are on the $y$-axis

Therefore, the equation of the hyperbola is of the form $a_{2}y_{2} −b_{2}x_{2} =1$

Since the vertices are $(0,±5)⇒a=5$

And the foci are $(0,±8)⇒ae=8$

We know that $b_{2}=a_{2}(e_{2}−1)=a_{2}e_{2}−a_{2}=64−25=39$

Thus the equation of the hyperbola is $25y_{2} −39x_{2} =1$