Hyperbola
Tangent to Hyperbola
Grade 11
Question:
<p>Given: \(4y^2 = x^2 + 1\). So, \(\left(\tan\theta, \frac{1}{2}\sec\theta\right)\) lies on hyperbola. The tangent at this point meets the co-ordinate axes at points \(A\) and \(B\). If the mid-point of \(AB\) is \((h, k)\), then the locus of the mid-point is:</p>
<p>\(\frac{1}{16k^2} - \frac{1}{4h^2} = 1\)</p>
<p>\(\frac{1}{16k^2} - \frac{1}{4h^2} = -1\)</p>
<p>\(x^2 - 4y^2 - 16x^2y^2 = 0\)</p>
<p>\(\frac{1}{4y^2} - \frac{1}{2x^2} = 4\)</p>
Step-by-Step Solution
Key Concept: Find the equation of the tangent to the hyperbola at the given point, determine where it meets the coordinate axes, find the midpoint of AB in terms of the parameter θ, then eliminate θ to get the locus equation.
<p><strong>Step 1:</strong> Verify the point lies on the hyperbola. The hyperbola is 4y² - x² = 1. At point P(tan θ, ½sec θ): 4(¼sec²θ) - tan²θ = sec²θ - tan²θ = 1 ✓</p><p><strong>Step 2:</strong> Find the tangent line equation. For hyperbola 4y² - x² = 1, using implicit differentiation: 8y(dy/dx) - 2x = 0, so dy/dx = x/(4y). At P(tan θ, ½sec θ): slope = tan θ/(2sec θ) = sin θ cos θ/2.</p><p><strong>Step 3:</strong> Write tangent equation: y - ½sec θ = (sin θ cos θ/2)(x - tan θ). Simplifying: y = (sin θ cos θ/2)x - sin²θ/2 + ½sec θ = (sin θ cos θ/2)x + cos²θ/(2cos θ).</p><p><strong>Step 4:</strong> Find point A (y-intercept, x = 0): y = cos²θ/(2cos θ) = cos θ/2. So A = (0, cos θ/2).</p><p><strong>Step 5:</strong> Find point B (x-intercept, y = 0): 0 = (sin θ cos θ/2)x + cos²θ/(2cos θ). Solving: x = -cos θ/sin θ = -cot θ. So B = (-cot θ, 0).</p><p><strong>Step 6:</strong> Midpoint AB: h = -cot θ/2, k = cos θ/4. From these: cot θ = -2h and cos θ = 4k.</p><p><strong>Step 7:</strong> Use cot²θ + 1 = csc²θ and cos²θ + sin²θ = 1. From cot θ = -2h: csc²θ = 1 + 4h². From cos θ = 4k: sin²θ = 1 - 16k². Also csc²θ = 1/sin²θ.</p><p><strong>Step 8:</strong> Therefore: 1 + 4h² = 1/(1 - 16k²). Cross-multiply: (1 + 4h²)(1 - 16k²) = 1. Expanding: 1 - 16k² + 4h² - 64h²k² = 1. Simplifying: 4h² - 16k² - 64h²k² = 0. Dividing by 4: h² - 4k² - 16h²k² = 0 or equivalently x² - 4y² - 16x²y² = 0.</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C