Chapter 11.1, Problem 56E

Reflecting Telescope The Hale telescope at the Mount Palomar Observatory has a 200-in. mirror, as shown in the figure. The mirror is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is 3.79 in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.

To determine

To find: The focal length of the parabolic mirror, that is the distance from the vertex to the focus.

Answer to Problem 56E

The focal length of the parabolic mirror is $659.63\text{-in}$ from the vertex.

Explanation of Solution

Definition used:

 “The equation of the parabola with vertex $V(0,0)$, focus $F\left(0,p\right)$ and directrix $y=-p$ is $x^2=4py$.”

Calculation:

From the figure observe that the parabola is open upward, hence assume the equation of the parabola be $x^2=4py$.

Since the reflector is $3.79$ in. deep and $200$ in. wide, the axis divides the $200$ in. wide in to two halves.

Therefore, the point $\left(0,3.79\right)$ on the axis (positive y-axis).

Let the diameter passing through $\left(0,3.79\right)$ meets the parabola $x^2=4py$ at $A\left(-100,3.79\right)$ and $B\left(100,3.79\right)$ as shown in Figure 1 and satisfies the equation $x^2=4py$.

Substitute $x=100\text{and}y=3.79$ in the equation $x^2=4py$,

$\begin{array}{l}{100}^2=4p\left(3.79\right)\\ 10000=15.16p\\ p=\frac{10000}{15.16}\\ p=659.63\end{array}$

Therefore, focus of the parabolic reflector is $F\left(0,659.63\right)$

Therefore, the focal length of the parabolic mirror is $659.63\text{-in}$ from the vertex.