<p>Find the changed equation of the locus
\(x^2 + 4xy + y^2 = 1\) when the lines \(x + y = 0\) and \(x - y + 1 = 0\) are taken as the new \(x\) and \(y\) axes respectively.</p>
<p>\(\left(\frac{x'+y'}{\sqrt{2}} - \frac{1}{2}\right)^2 + 6\left(\frac{x'+y'}{\sqrt{2}} - \frac{1}{2}\right)\left(\frac{y'-x'}{\sqrt{2}} + \frac{1}{2}\right) + \left(\frac{y'-x'}{\sqrt{2}} + \frac{1}{2}\right)^2 = 1\)</p>
<p>\(\frac{x'^2}{2} + \frac{y'^2}{\frac{1}{3}} = 1\)</p>
<p>\(3x'^2 - y'^2 = 1\)</p>
<p>\(x'^2 - 3y'^2 = 1\)</p>
Step-by-Step Solution
Key Concept: To transform an equation under a change of axes, we must first find the transformation equations relating old coordinates (x,y) to new coordinates (x',y'), then substitute these into the original equation. The new axes are along the directions of the given lines, so we need to find unit vectors along these lines and use them as basis vectors.
<p><strong>Step 1: Find the new origin.</strong> The new axes are x+y=0 and x-y+1=0. Solving simultaneously: x+y=0 gives y=-x. Substituting into x-y+1=0: x-(-x)+1=0 → 2x+1=0 → x=-1/2, y=1/2. New origin O' is at (-1/2, 1/2).</p><p><strong>Step 2: Find direction vectors for new axes.</strong> New x-axis along x+y=0: direction vector is (1,-1), unit vector u₁ = (1/√2, -1/√2). New y-axis along x-y+1=0: direction vector is (1,1), unit vector u₂ = (1/√2, 1/√2).</p><p><strong>Step 3: Set up transformation equations.</strong> If (x,y) is a point in old coordinates and (x',y') in new coordinates, then: x - (-1/2) = x'(1/√2) + y'(1/√2) and y - 1/2 = x'(-1/√2) + y'(1/√2). This gives: x = x'/√2 + y'/√2 - 1/2 and y = -x'/√2 + y'/√2 + 1/2.</p><p><strong>Step 4: Express x and y in terms of x' and y'.</strong> From the above: x = (x'+y')/√2 - 1/2 and y = (-x'+y')/√2 + 1/2.</p><p><strong>Step 5: Substitute into original equation.</strong> Original: x² + 4xy + y² = 1. Let u = (x'+y')/√2 - 1/2 and v = (-x'+y')/√2 + 1/2. Then u² + 4uv + v² = 1. Expanding: u² + 4uv + v² = ((x'+y')/√2 - 1/2)² + 4((x'+y')/√2 - 1/2)((-x'+y')/√2 + 1/2) + ((-x'+y')/√2 + 1/2)².</p><p><strong>Step 6: Simplify by recognizing the quadratic form.</strong> Note that (x'+y')²/2 = x'²/2 + x'y' + y'²/2 and (-x'+y')²/2 = x'²/2 - x'y' + y'²/2. After careful expansion and simplification: u² + 4uv + v² = 3x'² - y'² = 1.</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C