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: Find the slope of the tangent line to the parabola at \( (4, 6) \):
The equation of the parabola is given by: \[ y^2 = 9x \] which represents a rightward-opening parabola.
Differentiate the equation of the parabola \( y^2 = 9x \) with respect to \( x \): \[ 2y \frac{dy}{dx} = 9 \quad \Rightarrow \quad \frac{dy}{dx} = \frac{9}{2y} \] Substituting \( y = 6 \): \[ \frac{dy}{dx} = \frac{9}{2 \times 6} = \frac{3}{4} \] So, the slope of the tangent at \( (4, 6) \) is \( \frac{3}{4} \).
Step 2: Equation of the tangent line:
The equation of the tangent line at \( (4, 6) \) can be written as: \[ y - 6 = \frac{3}{4} (x - 4) \] Simplifying: \[ y = \frac{3}{4}x + 3 \]
Step 3: Find the equation of the circle:
The circle touches the parabola at \( (4, 6) \) and the positive x-axis, meaning the center of the circle lies on the x-axis.
The center of the circle is at \( (4, r) \), where \( r \) is the radius of the circle, and the equation of the circle is: \[ (x - 4)^2 + (y - r)^2 = r^2 \]
Step 4: Substitute the point \( (4, 6) \) into the equation of the circle:
Substituting \( x = 4 \) and \( y = 6 \) into the equation: \[ (4 - 4)^2 + (6 - r)^2 = r^2 \] Simplifying: \[ (6 - r)^2 = r^2 \quad \Rightarrow \quad 36 - 12r + r^2 = r^2 \] This simplifies to: \[ 36 = 12r \quad \Rightarrow \quad r = \frac{36}{12} = 3 \] Thus, the radius of the circle is \( 3 \).