<p>The family of lines \((3\sec\theta + 5\csc\theta)x + (7\sec\theta - 3\csc\theta)y + 11(\sec\theta - \csc\theta) = 0\) passes through a fixed point. Find the fixed point.</p>
Step-by-Step Solution
Key Concept: Rewrite the equation by grouping terms with sec θ and csc θ separately, then use the fact that for the equation to hold for all values of θ, the coefficients of sec θ and csc θ must independently equal zero at the fixed point.
<p><strong>Step 1:</strong> Rewrite the equation by grouping sec θ and csc θ terms:</p><p>sec θ(3x + 7y + 11) + csc θ(-5x - 3y - 11) = 0</p><p><strong>Step 2:</strong> For this equation to be satisfied for all values of θ (i.e., for all values of sec θ and csc θ), both coefficients must equal zero independently:</p><p>• Coefficient of sec θ: 3x + 7y + 11 = 0 ... (1)</p><p>• Coefficient of csc θ: -5x - 3y - 11 = 0, or 5x + 3y + 11 = 0 ... (2)</p><p><strong>Step 3:</strong> Solve the system of equations (1) and (2):</p><p>From (1): 3x + 7y = -11</p><p>From (2): 5x + 3y = -11</p><p>Multiply (1) by 5: 15x + 35y = -55</p><p>Multiply (2) by 3: 15x + 9y = -33</p><p>Subtract: 26y = -22, so y = -11/13</p><p>Substitute into (2): 5x + 3(-11/13) = -11</p><p>5x - 33/13 = -11</p><p>5x = -11 + 33/13 = (-143 + 33)/13 = -110/13</p><p>x = -22/13</p><p><strong>Step 4:</strong> Verify by substitution into the original equation that the fixed point is (-22/13, -11/13).</p><p>∴ Answer: A (Fixed point is (-22/13, -11/13))</p>
Correct Answer: A