(a)

In a searchlight, the bulb is placed at the focus of a parabolic mirror  from the vertex.

The point of vertex is .

The focal diameter of the bulb is .

Since the focal diameter diameter of the bulb is , is lie on positive side of -axis and  is lie on negative side of -axis.

The points lie on the parabola are and .

Vertex of the parabola is .

The equation of parabola is .

Substitute and in the above eqaution.

Substitute and in .

Equation of parabola is .

(b)

The focal diameter of the bulb is .

Since the focal diameter diameter of the bulb is , is lie on positive side of -axis and  is lie on negative side of -axis.

The points lie on the parabola are and .

Vertex of the parabola is .

Substitute and in .

Substitute and in .

Equation of parabola is . 

The depth of both searchlights is  feet.

The depth represents the distance the reflector spans horizontally.

The width represents the distance the reflector spans vertically.

Find the width of the first searchlight.

Substitute in .

.

Width of the first searchlight is and .

Here width lie on positive side of -axis i.e and negative side of -axis i.e .

Total width of the search light is .

Find the width of the second searchlight.

Substitute in .

.

Width of the first searchlight is and .

Here width lie on positive side of -axis i.e and negative side of -axis i.e .

Total width of the search light is .

The Difference of the width of the search light is  .

Width of the opening light is much greater than the width of second light.

Width of the opening light is much greater than the width of second light.