<p>If \(y + 3x = 0\) is the equation of a chord of the circle \(x^2 + y^2 - 30x = 0\), then the equation of the circle with this chord as diameter is</p>
<p>\(x^2 + y^2 + 3x + 9y = 0\)</p>
<p>\(x^2 + y^2 - 3x + 9y = 0\)</p>
<p>\(x^2 + y^2 - 3x - 9y = 0\)</p>
<p>\(x^2 + y^2 + 3x - 9y = 0\)</p>
Step-by-Step Solution
Key Concept: A chord becomes a diameter of a new circle; find the circle passing through the endpoints of the chord. The key is that any circle through the chord's endpoints with the chord as diameter must satisfy: the center lies on the perpendicular bisector of the chord, and the radius equals half the chord length.
<p><strong>Step 1:</strong> Rewrite the given circle: x² + y² - 30x = 0, so center is (15, 0) and radius² = 225.</p><p><strong>Step 2:</strong> Use the family of circles through the chord: (x² + y² - 30x) + λ(y + 3x) = 0, or x² + y² + 3λx + λy - 30x = 0.</p><p><strong>Step 3:</strong> For a circle with the chord as diameter, the original center (15, 0) must lie on the perpendicular bisector. Alternatively, rewrite as: x² + y² + (3λ - 30)x + λy = 0.</p><p><strong>Step 4:</strong> The center is at (-(3λ - 30)/2, -λ/2). For the chord y + 3x = 0 to be a diameter, the center must satisfy the perpendicular bisector condition. The chord has slope -3, so perpendicular has slope 1/3.</p><p><strong>Step 5:</strong> The perpendicular bisector passes through (15, 0) with slope 1/3: y - 0 = (1/3)(x - 15), giving 3y = x - 15, or x - 3y - 15 = 0.</p><p><strong>Step 6:</strong> Substitute center coordinates: (-(3λ - 30)/2) - 3(-λ/2) - 15 = 0 → -(3λ - 30)/2 + 3λ/2 - 15 = 0 → (-3λ + 30 + 3λ)/2 - 15 = 0 → 30/2 - 15 = 0 ✓ (check: use λ = 5).</p><p><strong>Step 7:</strong> With λ = 5: x² + y² - 15x + 5y = 0, which simplifies to x² + y² - 15x + 5y = 0.</p><p>∴ Answer: C</p>
Correct Answer: C