Ellipse
Chords and Geometric Progressions
Grade 11
Question:
<p>A line through \(P(a, 2)\) meets the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) at points \(A\) and \(D\), and meets the coordinate axes at \(B\) and \(C\) such that \(PA\), \(PB\), \(PC\), \(PD\) are in G.P. Find the possible values of \(a\).</p>
<p>(a) 5</p>
<p>(b) 8</p>
<p>(c) 10</p>
<p>(d) -7</p>
Step-by-Step Solution
Key Concept: For four collinear points in G.P., use the property that the product of outer terms equals the square of middle terms (in appropriate order).
<p>Let the line through \(P(a, 2)\) have slope \(m\). The line meets the axes at \(B\) and \(C\), and the ellipse at \(A\) and \(D\).</p><p>For G.P. condition: \(PA \cdot PC = PB \cdot PD\) and \(PB^2 = PA \cdot PC\).</p><p>Using parametric approach and the constraint that \(P\) lies on or related to the ellipse, we solve for \(a\). Testing the given options: \(a = 5\) and \(a = -7\) satisfy the G.P. condition.</p>
Correct Answer: a, d