Circles
Circle Equation with Diameter
Grade 11
Question:
<p>Find the equation of the circle circumscribing triangle <i>DAO</i> with diameter <i>AE</i>, where <i>A(a,0)</i> and <i>E\left(0, \frac{3a}{4}\right)</i>.</p>
<p>(A) <i>x^2 + y^2 - 7x + \frac{21y}{4} = 0</i></p>
<p>(B) <i>4(x^2 + y^2 - 7x) - 21y = 0</i></p>
<p>(C) <i>x^2 + y^2 - 7x - \frac{21y}{4} = 0</i></p>
<p>(D) <i>4x^2 + 4y^2 - 28x + 21y = 0</i></p>
Step-by-Step Solution
Key Concept: Use the diameter form of a circle equation: the locus of points P such that ∠APE = 90°.
<p><strong>Step 1:</strong> The circle with diameter <i>AE</i> has the equation:</p><p>\[(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0\]</p><p><strong>Step 2:</strong> Substituting <i>A(a,0)</i> and <i>E\left(0, \frac{3a}{4}\right)</i>:</p><p>\[x(x - a) + y\left(y - \frac{3a}{4}\right) = 0\]</p><p>\[x^2 - ax + y^2 - \frac{3a}{4}y = 0\]</p><p><strong>Step 3:</strong> With <i>a = 7</i>:</p><p>\[x^2 + y^2 - 7x - \frac{21y}{4} = 0\]</p><p><strong>Step 4:</strong> Multiplying by 4:</p><p>\[4(x^2 + y^2 - 7x) - 21y = 0\]</p><p>∴ Answer is <i>(B)</i>.</p>
Correct Answer: B