Solve the problem.
-For a hyperbolic mirror the two foci are $44 \mathrm {~cm}$ apart. The distance of the vertex from one focus is 10 $\mathrm { cm }$ and from the other focus is $34 \mathrm {~cm}$. 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.
A) $\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 340 } = 1$
B) $\frac { y ^ { 2 } } { 144 } - \frac { x ^ { 2 } } { 340 } = 1$
C) $\frac { y ^ { 2 } } { 144 } - \frac { x ^ { 2 } } { 340 } = 5$
D) $\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 340 } = 5$

Question

Quizplus · Accepted Answer

The distance between the two foci is $44 \mathrm{cm}$, so the distance from the center to either focus (denoted as $c$) is $22 \mathrm{cm}$. The distance from the vertex to one focus is $10 \mathrm{cm}$ and to the other focus is $34 \mathrm{cm}$, which means the distance from the center to a vertex (denoted as $a$) is $12 \mathrm{cm}$ (half the difference between $34 \mathrm{cm}$ and $10 \mathrm{cm}$). The relationship $c^2 = a^2 + b^2$ (for hyperbolas) allows us to find $b^2$. Here, $c = 22$, $a = 12$, so $b^2 = c^2 - a^2 = 22^2 - 12^2 = 340$. Since the foci are on the $y$-axis, the equation of the hyperbola is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, which simplifies to $\frac{y^2}{144} - \frac{x^2}{340} = 1$.

Solve the Problem 44 cm44 \mathrm {~cm}44 cm Apart the Distance of the Vertex from One Focus Is

Solve the Problem $44 \mathrm {~cm}$ Apart the Distance of the Vertex from One Focus Is