Question 63 of 70intermediate🔧 ApplyLong Answer4 marks

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

Correct Answer

Let the vertex of the parabolic reflector be at the origin and its axis along the x-axis. Then the equation of the parabola is y2=4axy^2 = 4ax, where aa is the distance of the focus from the vertex.

The depth of the reflector is 5 cm, so the rim lies on the line x=5x = 5. The diameter is 20 cm, hence the radius is 10 cm. Therefore, the point (5,10)(5, 10) lies on the parabola.

Substituting the coordinates into the equation:

(10)2=4a(5)(10)^2 = 4a(5) ⇒100=20a\Rightarrow 100 = 20a ⇒a=5\Rightarrow a = 5

Thus, the focus is at (a,0)=(5,0)(a, 0) = (5, 0). Hence, the focus is at a distance of 5 cm from the vertex along the axis.

Exercise: Miscellaneous Exercise on Chapter 10 | Q: 1 | (Page 29)
For More Understanding

Explanation

The problem is solved by modeling the parabolic reflector as a parabola with vertex at the origin and axis along the x-axis. The standard equation y2=4axy^2 = 4ax is used, where aa is the focal length.

The given depth gives the x-coordinate of the rim, and half the diameter gives the y-coordinate. Substituting yields a=5a = 5 cm, so the focus is 5 cm from the vertex.

This matches the method shown in Example 17 of the textbook.

Solution Steps

  1. Step 1: Assume vertex at origin, axis along x-axis, equation y2=4axy^2 = 4ax.

  2. Step 2: Depth = 5 cm → x=5x = 5; radius = 10 cm → y=10y = 10; point (5,10)(5, 10) on parabola.

  3. Step 3: Substitute: 102=4a×5→100=20a→a=510^2 = 4a \times 5 \rightarrow 100 = 20a \rightarrow a = 5.

  4. Step 4: Focus is at (a,0)=(5,0)(a, 0) = (5, 0). Distance from vertex = 5 cm.

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