Parabola
Circle passing through focal chord
Grade 11
Question:
<p>If a circle with center K(<i>c</i>, 0) passes through two points P and Q on the parabola <i>y</i><sup>2</sup> = 4<i>x</i>, and satisfies the condition (1/(PK)<sup>2</sup>) + (1/(QK)<sup>2</sup>) is independent of the angle θ, find the value of <i>c</i>.</p>
Step-by-Step Solution
Key Concept: The condition for independence from angle θ in the reciprocal sum of squares of focal radii determines the center location uniquely.
<p><strong>Step 1:</strong> Points on parabola in parametric form: P and Q lie on <i>y</i><sup>2</sup> = 4<i>x</i></p><p><strong>Step 2:</strong> Circle with center K(<i>c</i>, 0) passes through P and Q, so |PK| = |QK| = radius</p><p><strong>Step 3:</strong> Using the focal chord property with center at K: (1/r<sub>1</sub><sup>2</sup>) + (1/r<sub>2</sub><sup>2</sup>)</p><p><strong>Step 4:</strong> For this expression to be independent of θ: (16cos<sup>2</sup>θ + 8c sin<sup>2</sup>θ)/(16c<sup>2</sup>) = constant</p><p><strong>Step 5:</strong> This requires 8<i>c</i> = 16, thus <i>c</i> = 2</p>
Correct Answer: 2