Question

For a hyperbolic mirror the two foci are 42 cm apart. The distance of the vertex from one focus is 6 cm and from the other focus is 36 cm. Position a coordinate system with the origin at the center of the hyperbola and with the foci on the y-axis. Find the equation of the hyperbola.

Answer 1

`(y^2)/(225) -(x^2)/(216)=1`

Explanation:

For a hyperbolic mirror the two foci are 42 cm apart.

The distance between the foci = 2c.

Therefore:

• 2c=42
• c=21

The distance of the vertex from one focus = 6 cm

The distance of the vertex from the other focus = 36 cm

2a=36-6=30

• a=15

Now:

`c^2=a^2+b^2\\21^2=15^2+b^2\\b^2=21^2-15^2\\b^2=216\\b=6√(6)`

If the transverse axis lies on the y-axis, and the hyperbola is centered at the origin. Then the hyperbola has an equation of the form:

`(y^2)/(a^2) -(x^2)/(b^2)=1`

Therefore, the equation of the hyperbola is:

`(y^2)/(225) -(x^2)/(216)=1`