<p>Two chords are drawn from the point <i>P</i>(<i>h</i>, <i>k</i>) on the circle <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = <i>hx</i> + <i>ky</i>. If the <i>y</i>-axis divides both the chords in the ratio 2:3, then which of the following may be correct?</p>
<p>(a) \(k^2 > 15h^2\)</p>
<p>(b) \(15k^2 > h^2\)</p>
<p>(c) \(h^2 = 15k^2\)</p>
<p>(d) \(k^2 > 5h^2\)</p>
Step-by-Step Solution
Key Concept: If the y-axis divides a chord joining points A and B in ratio 2:3, then the y-axis intersects the chord at a point that divides it internally. Using section formula and the constraint that both chords from P(h,k) satisfy this condition leads to a relationship between h and k.
<p><strong>Step 1: Rewrite the circle equation</strong></p><p>The circle is x² + y² = hx + ky, which can be rewritten as:</p><p>(x - h/2)² + (y - k/2)² = (h² + k²)/4</p><p>Center: C(h/2, k/2), Radius: √(h² + k²)/2</p><p>Note: P(h, k) lies on the circle since h² + k² = h·h + k·k ✓</p><p><strong>Step 2: Set up the chord division condition</strong></p><p>Let a chord from P(h, k) meet the circle at point Q(x₁, y₁). The y-axis divides PQ in ratio 2:3.</p><p>If the y-axis meets PQ at point M(0, y₀), then by section formula:</p><p>M divides PQ in ratio 2:3, so: M = (3P + 2Q)/5</p><p>Since M lies on y-axis: 0 = (3h + 2x₁)/5</p><p>Therefore: x₁ = -3h/2</p><p><strong>Step 3: Apply the circle constraint</strong></p><p>Since Q(x₁, y₁) lies on the circle:</p><p>x₁² + y₁² = hx₁ + ky₁</p><p>Substituting x₁ = -3h/2:</p><p>9h²/4 + y₁² = h(-3h/2) + ky₁</p><p>9h²/4 + y₁² = -3h²/2 + ky₁</p><p>9h²/4 + 3h²/2 + y₁² = ky₁</p><p>15h²/4 + y₁² = ky₁</p><p>y₁² - ky₁ + 15h²/4 = 0</p><p><strong>Step 4: Condition for two real chords</strong></p><p>Since there are TWO distinct chords satisfying the condition, the quadratic in y₁ must have TWO distinct real roots.</p><p>For this: Discriminant > 0</p><p>k² - 4(15h²/4) > 0</p><p>k² - 15h² > 0</p><p>k² > 15h²</p><p><strong>∴ Answer:</strong> a</p>
Correct Answer: a