Question

If a parabolic reflector is 20cm in diameter and 5cm deep, and then the focus is
(A) (0,5)
(B) (5,0)
(C) (0,-5)
(D) (-5,0)

Answer 1

Hint: We measure the diameter and divide it by two to get radius.

The formula for parabola is:

${y^2} = 4ax$or ${x^2} = 4ay$

From the above equation we can find a and the focal point for the parabola is $(a,0)$ or $(0, a).$

A focal point of a parabola is that point on the axis from which all points on the curve are equidistant.

Complete step-by-step answer:

First we draw a diagram of the question clearly depicting the values needed to solve the question.

Let we take the formula for parabola is

${y^2} = 4ax$

We need to find the focus so take point $A$ as shown in figure and the center of the parabola is $C$. The parabola is $5cm$ deep so the $OC = 5cm$ and the diameter is 20 cm so radius will be $10cm$ so the coordinate of $A$ is $(5,10)$. That is on the parabola it will satisfy the equation of parabola and the focal point of parabola is $(a,0)$.

Now put the value of $y=10$ and $x=5$ in the equation of parabola.

{y^2} = 4ax \\\

On substituting the $y, x$ values,

$\Rightarrow$ 100 = 4 \times a \times 5 \\\

On simplifying the above equation, we get

$\Rightarrow$ a = \dfrac{{100}}{{4 \times 5}} \\\

$\Rightarrow$ $a = 5 \\\$

$\therefore$ The focus point becomes (5,0). So, option B is correct.

Note:

It is recommended to plot a graph for the questions that will give an idea of the exact conditions of the problem. Some students skip graphs and this may result in errors in the final answer. The focal point is dependent on that parabola on which the axis is symmetrical.