Straight Lines
Rotation of Axes / Pair of Straight Lines
Grade 11
Question:
<p>The equation of a pair of straight lines is \(ax^2 + 2hxy + by^2 = 0\). By what angle must the axes be rotated so that the term containing \(xy\) in the equation may be removed?</p>
<p>\(\tan^{-1}\left(\frac{2h}{a-b}\right)\)</p>
<p>\(\frac{1}{2}\tan^{-1}\left(\frac{2h}{a-b}\right)\)</p>
<p>\(\tan^{-1}\left(\frac{h}{a-b}\right)\)</p>
<p>\(\frac{1}{2}\tan^{-1}\left(\frac{h}{a-b}\right)\)</p>
Step-by-Step Solution
Key Concept: The xy term vanishes after rotation when the new axes align with the principal directions of the conic. This occurs when the axes are rotated by angle θ where tan(2θ) = 2h/(a-b), eliminating the mixed product term.
<p><strong>Step 1:</strong> For a general second-degree equation ax² + 2hxy + by² = 0, when axes are rotated by angle θ, the new coordinates are related by:</p><p>x = X cos θ - Y sin θ</p><p>y = X sin θ + Y cos θ</p><p><strong>Step 2:</strong> Substituting these in the original equation and simplifying, the coefficient of XY term becomes:</p><p>2h' = 2h cos(2θ) + (b - a) sin(2θ)</p><p><strong>Step 3:</strong> To remove the XY term, set 2h' = 0:</p><p>2h cos(2θ) + (b - a) sin(2θ) = 0</p><p>2h cos(2θ) = (a - b) sin(2θ)</p><p><strong>Step 4:</strong> Dividing both sides by cos(2θ):</p><p><strong>tan(2θ) = 2h/(a - b)</strong></p><p>Therefore: <strong>θ = (1/2) tan⁻¹[2h/(a - b)]</strong></p><p>∴ Answer: B</p>
Correct Answer: B