Question

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.

Short answer

The focal length of parabola is $659.83$inch

Step-by-step solution

Step 1. Given information

A parabola formed by cables is given as-

Step 2. Concept used

The equation of a parabola having a vertex at origin is –

$x^2=4py$

Where, $p>0$ then parabola open upwards

$p<0$ then parabola open downwards

The parabolic part of cables are vertical so the equation is of the form $x^2=4py$and focus is $(0,p)$

We can find the coordinates of endpoints i.e., topmost points of the parabolic part by using the distance between the two towers.

Then, on substituting any one endpoint in the parabolic equation, to get the value for $4p$

Then, again substituting the value of $4p$in the parabolic equation $x^2=4py$to get the equation.

Step 3. Calculation

The parabolic part of cables has an equation of the form -

$x^2=4py$

The focus is at $(0,p)$.

The vertex is at $(0,0)$.

The mirror is $3.79$inch deep and $200$ inch width, which means endpoints of the parabolic shape has the coordinate –

$(100,3.79)\&(-100,3.79)$

Substituting, the point $(100,3.79)$ in the equation of the parabola $x^2=4py$, we get –

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

The focus is at $(0,659.63)$ and \[p=659.63\]then the distance between focus and vertex is $659.63$ inch.