<p>Circle touches the <em>x</em>-axis at (3, 0) and making an intercept of length 8 on the <em>y</em>-axis. The equation of the circle is:</p>
<p>(A) \(x^2 + y^2 - 6x - 10y + 9 = 0\)</p>
<p>(B) \(x^2 + y^2 - 6x + 10y + 9 = 0\)</p>
<p>(C) \(x^2 + y^2 + 6x - 10y + 9 = 0\)</p>
<p>(D) \(x^2 + y^2 + 6x + 10y + 9 = 0\)</p>
Step-by-Step Solution
Key Concept: If a circle touches the x-axis at point (3,0), its center must be at (3,r) where r is the radius. Use the chord-intercept condition on the y-axis to find r.
<p><strong>Step 1:</strong> Since the circle touches the x-axis at (3,0), the center is at (3,r) where r is the radius (the center lies on the perpendicular to x-axis at the point of tangency).</p><p><strong>Step 2:</strong> The circle makes an intercept of 8 on the y-axis. The distance from center (3,r) to the y-axis is 3. Using the chord-intercept formula: if a chord of length 2l is at distance d from center with radius R, then d² + l² = R². Here, d = 3, 2l = 8 (so l = 4), and R = r.</p><p><strong>Step 3:</strong> Applying the formula: 3² + 4² = r² → 9 + 16 = r² → r² = 25 → r = 5</p><p><strong>Step 4:</strong> Center is at (3,5) and radius is 5. The equation is (x-3)² + (y-5)² = 25, which expands to x² + y² - 6x - 10y + 9 = 0</p><p><strong>Verification:</strong> Distance from (3,5) to x-axis = 5 ✓. Distance from (3,5) to y-axis = 3, and √(25-9) = 4, so intercept = 8 ✓</p><p>∴ Answer: A</p>
Correct Answer: A