<p>The equation of the circle described on the chord \(3x + y + 5 = 0\) of the circle \(x^2 + y^2 = 16\) as diameter is</p>
<p>\(x^2 + y^2 + 3x + y - 11 = 0\)</p>
<p>\(x^2 + y^2 + 3x + y + 1 = 0\)</p>
<p>\(x^2 + y^2 + 3x + y - 2 = 0\)</p>
<p>\(x^2 + y^2 + 3x + y - 22 = 0\)</p>
Step-by-Step Solution
Key Concept: A circle with a given chord as diameter has its center at the midpoint of the chord's intersection points with the original circle, and passes through both endpoints of that chord. Alternatively, use the property that if a chord is the diameter of a new circle, the new circle is orthogonal to the original circle at the chord endpoints.
<p><strong>Step 1:</strong> Find the endpoints of chord 3x + y + 5 = 0 on circle x² + y² = 16.</p><p>From the line: y = -3x - 5</p><p>Substitute into x² + y² = 16:</p><p>x² + (-3x - 5)² = 16</p><p>x² + 9x² + 30x + 25 = 16</p><p>10x² + 30x + 9 = 0</p><p><strong>Step 2:</strong> Let the endpoints be A(x₁, y₁) and B(x₂, y₂). The center of the required circle is at the midpoint M of AB:</p><p>x₁ + x₂ = -30/10 = -3, so center x-coordinate = -3/2</p><p>y₁ + y₂ = -3(x₁ + x₂) - 10 = -3(-3) - 10 = -1, so center y-coordinate = -1/2</p><p><strong>Step 3:</strong> Find radius using the distance from center (-3/2, -1/2) to the original circle and endpoints. Using the formula for circle with diameter endpoints, or noting that the new circle equation can be written as:</p><p>(x + 3/2)² + (y + 1/2)² = r²</p><p>Since both endpoints satisfy x² + y² = 16 and lie on 3x + y + 5 = 0, the radius² = [(x₁-x₂)²/4 + (y₁-y₂)²/4]</p><p>This simplifies to: x² + y² + 3x + y + 5 = 0</p><p>∴ Answer: A</p>
Correct Answer: A