<p>Find the locus of the middle points of the chords of the circle \(x^2 + y^2 = a^2\) which pass through a given point \((x_0, y_0)\).</p>
Step-by-Step Solution
Key Concept: Use the property that the line joining the center of a circle to the midpoint of any chord is perpendicular to that chord. For chords passing through a fixed point, the locus of midpoints lies on a circle whose diameter connects the center and the given fixed point.
<p><strong>Step 1:</strong> Let M(h,k) be the midpoint of a chord of circle x² + y² = a² that passes through point P(x₀,y₀).</p><p><strong>Step 2:</strong> The line from origin O to midpoint M(h,k) is perpendicular to the chord. Therefore, the slope of OM × slope of chord = -1.</p><p><strong>Step 3:</strong> Slope of OM = k/h. The chord passes through M(h,k) and P(x₀,y₀), so slope of chord = (y₀-k)/(x₀-h).</p><p><strong>Step 4:</strong> Apply perpendicularity: (k/h) · ((y₀-k)/(x₀-h)) = -1</p><p><strong>Step 5:</strong> Simplifying: k(y₀-k) = -h(x₀-h)</p><p>k·y₀ - k² = -h·x₀ + h²</p><p>h² + k² = h·x₀ + k·y₀</p><p><strong>Step 6:</strong> Replacing (h,k) with (x,y): <strong>x² + y² - x·x₀ - y·y₀ = 0</strong></p><p>This is a circle passing through origin with diameter along OP.</p>
Correct Answer: x^2 + y^2 - x*x0 - y*y0 = 0