Question

A circle touches the parabola $ y^2 = 9x $ at $ (4, 6) $ and the positive x-axis. Find the radius of the circle.

• A. \( \frac{20}{3} \)
• B. \( \frac{10}{3} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{5}{3} \)

Hint:

For problems involving a circle and a parabola, first find the slope of the tangent to the parabola and use it to form the equation of the tangent line. Then, use the fact that the circle touches the tangent to find the radius.

Answer 1

The Correct Option is B

Step 1: Determine the slope of the tangent line to the parabola at \( (4, 6) \):
The given parabola equation is \( y^2 = 9x \).
Differentiating implicitly with respect to \( x \):
\[2y \frac{dy}{dx} = 9 \quad \Rightarrow \quad \frac{dy}{dx} = \frac{9}{2y}\]
At the point \( (4, 6) \), substitute \( y = 6 \):
\[\frac{dy}{dx} = \frac{9}{2 \times 6} = \frac{3}{4}\]
The slope of the tangent line at \( (4, 6) \) is \( \frac{3}{4} \).

Step 2: Formulate the equation of the tangent line:
Using the point-slope form with \( (4, 6) \) and slope \( \frac{3}{4} \):
\[y - 6 = \frac{3}{4} (x - 4)\]
Simplifying yields:
\[y = \frac{3}{4}x + 3\]

Step 3: Derive the equation of the circle:
The circle is tangent to the parabola at \( (4, 6) \) and the positive x-axis. This implies the circle's center lies on the x-axis.
Let the center be \( (4, r) \), where \( r \) is the radius. The equation of the circle is:
\[(x - 4)^2 + (y - r)^2 = r^2\]

Step 4: Substitute the point \( (4, 6) \) into the circle's equation:
Substitute \( x = 4 \) and \( y = 6 \) into the equation:
\[(4 - 4)^2 + (6 - r)^2 = r^2\]
Simplify:
\[0^2 + (6 - r)^2 = r^2 \quad \Rightarrow \quad (6 - r)^2 = r^2\]
Expand and simplify:
\[36 - 12r + r^2 = r^2\]
\[36 = 12r \quad \Rightarrow \quad r = \frac{36}{12} = 3\]
The radius of the circle is \( 3 \).