Straight Lines
Pair of Lines
Grade 11
Question:
<p>A pair of lines is given by \(ax^2 + 2hxy + by^2 = 0\). If point \((x_1, y_1)\) lies on one of the lines, find the relationship between the coefficients \(a\), \(b\), and \(h\).</p>
<p>(A) \(8h^2 = 9ab\)</p>
<p>(B) \(9ab = 8h^2\)</p>
<p>(C) \(4h^2 = 9ab\)</p>
<p>(D) \(ab = 8h^2\)</p>
Step-by-Step Solution
Key Concept: For a pair of lines through the origin, use the condition that specific points create proportional distances.
<p><strong>Step 1:</strong> For pair of lines, if \((x_1, y_1)\) lies on one line, substitute in \(ax^2 + 2hxy + by^2 = 0\)</p><p><strong>Step 2:</strong> The two slopes at point \((x_1, 0)\) satisfy: \(by^2 + 2hx_1y + ax_1^2 = 0\)</p><p><strong>Step 3:</strong> From the condition that \(C(x_1, 2y_1)\), \(B(x_1, y_1)\), \(A(x_1, 0)\) form configuration:</p><p><strong>Step 4:</strong> \(\frac{9y_1^2}{2y_1^2} = \frac{4h^2x_1^2 + ab}{ab}\)</p><p><strong>Step 5:</strong> Simplifying: \(\frac{9}{2} = \frac{4h^2}{ab}\)</p><p>∴ \(9ab = 8h^2\)</p>
Correct Answer: B