Question

A circle touches the \( x \)-axis at \( (9,0) \). If it also touches the straight line \( y = 14 \), then the equation of the circle is

• A. \( (x-9)^2 + (y-7)^2 = 49 \)
• B. \( x^2 + (y-7)^2 = 49 \)
• C. \( (x-9)^2 + y^2 = 49 \)
• D. \( (x-9)^2 + (y-7)^2 = 81 \)
• E. \( (x-7)^2 + (y-9)^2 = 49 \)

Hint:

If a circle touches two parallel lines, center lies midway and radius is half the distance.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The equation of a circle with center (h, k) and radius r is \((x-h)^2 + (y-k)^2 = r^2\). The key to this problem is using the tangency conditions to find the center (h, k) and the radius r.
Step 2: Key Formula or Approach:
1. If a circle touches the x-axis at a point (h, 0), its center must be at (h, k) and its radius must be \(r = |k|\). 2. The distance from the center of the circle to any tangent line is equal to the radius.
Step 3: Detailed Explanation:
1. Find the center's x-coordinate and relate radius to the y-coordinate. The circle touches the x-axis at (9, 0). This means the x-coordinate of the center is \(h = 9\). The center is at \((9, k)\). The radius `r` is the distance from the center (9, k) to the point of tangency (9, 0), so \(r = |k - 0| = |k|\). Since the circle must be above the x-axis to also touch y=14, we can say \(k>0\), so \(r = k\). 2. Use the second tangency condition. The circle also touches the line \(y = 14\). The distance from the center \((9, k)\) to the horizontal line \(y = 14\) must also be equal to the radius `r`. This distance is given by \(|k - 14|\). So, we have \(r = |k - 14|\). 3. Solve for k and r. We now have two expressions for the radius: \(r = k\) and \(r = |k - 14|\). Equating them gives \(k = |k - 14|\). This single equation gives two possibilities: a) \(k = k - 14\), which simplifies to \(0 = -14\), an impossibility. b) \(k = -(k - 14)\), which simplifies to \(k = -k + 14\). Solving for k: \[ 2k = 14 \] \[ k = 7 \] So, the center of the circle is \((h, k) = (9, 7)\). The radius is \(r = k = 7\). 4. Write the equation of the circle. Using the standard form with center (9, 7) and radius 7: \[ (x - 9)^2 + (y - 7)^2 = 7^2 \] \[ (x - 9)^2 + (y - 7)^2 = 49 \] Step 4: Final Answer:
The equation of the circle is \((x-9)^2 + (y - 7)^2 = 49\).