<p>The line \(x + 3y = 0\) is a diameter of the circle \(x^2 + y^2 - 6x + 2y = 0\).</p><p><em>State whether the statement is true or false.</em></p>
Step-by-Step Solution
Key Concept: A line is a diameter of a circle if and only if the center of the circle lies on that line. Find the circle's center by rewriting the equation in standard form, then check if the center satisfies the line equation.
<p><strong>Step 1:</strong> Convert the circle equation to standard form by completing the square.</p><p>Given: $x^2 + y^2 - 6x + 2y = 0$</p><p>$(x^2 - 6x) + (y^2 + 2y) = 0$</p><p>$(x^2 - 6x + 9) + (y^2 + 2y + 1) = 9 + 1$</p><p>$(x - 3)^2 + (y + 1)^2 = 10$</p><p><strong>Step 2:</strong> Identify the center of the circle.</p><p>Center = $(3, -1)$</p><p><strong>Step 3:</strong> Check if the center lies on the line $x + 3y = 0$.</p><p>Substitute $(3, -1)$: $3 + 3(-1) = 3 - 3 = 0$ ✓</p><p>The center lies on the line, so the line passes through the center.</p><p><strong>Step 4:</strong> Conclusion.</p><p>Since the line passes through the center of the circle, it is indeed a diameter of the circle.</p><p>∴ The statement is <strong>TRUE</strong></p>
Correct Answer: B