Circles
Chord as diameter
Grade 11
Question:
<p>The intercept on the line <em>y</em> = <em>x</em> by the circle <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> − 2<em>x</em> = 0 is <em>AB</em>. Equation of the circle on <em>AB</em> as a diameter is</p>
<p>\(x^2 + y^2 - x - y = 0\)</p>
<p>\(x^2 + y^2 - x + y = 0\)</p>
<p>\(x^2 + y^2 + x + y = 0\)</p>
<p>\(x^2 + y^2 + x - y = 0\)</p>
Step-by-Step Solution
Key Concept: Find the intersection points of the line with the circle, then use these two points as diameter endpoints. The circle with diameter AB has equation (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0.
<p><strong>Step 1:</strong> Find intersection points of y = x with circle x² + y² − 2x = 0</p><p>Substitute y = x: x² + x² − 2x = 0 → 2x² − 2x = 0 → 2x(x − 1) = 0</p><p>So x = 0 or x = 1, giving points A(0, 0) and B(1, 1)</p><p><strong>Step 2:</strong> Use diameter form: if AB is diameter, then for any point P(x, y) on the circle:</p><p>(PA) · (PB) = 0 (dot product of position vectors from endpoints)</p><p>Vector PA = (x − 0, y − 0) and Vector PB = (x − 1, y − 1)</p><p>(x)(x − 1) + (y)(y − 1) = 0</p><p>x² − x + y² − y = 0</p><p><strong>Step 3:</strong> Verify: x² + y² − x − y = 0</p><p>∴ Answer: <strong>x² + y² − x − y = 0</strong></p>
Correct Answer: A