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

Answer:

$\dfrac{y^2}{225} -\dfrac{x^2}{216}=1$

Step-by-step 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:

$c^2=a^2+b^2\\21^2=15^2+b^2\\b^2=21^2-15^2\\b^2=216\\b=6\sqrt{6}$

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:

$\dfrac{y^2}{a^2} -\dfrac{x^2}{b^2}=1$

Therefore, the equation of the hyperbola is:

$\dfrac{y^2}{225} -\dfrac{x^2}{216}=1$