<p>If a tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), whose centre is C meets the major and minor axes produced at P and Q respectively then \(\frac{a^2}{CP^2} + \frac{b^2}{CQ^2}\) is equal to ___.</p>
Step-by-Step Solution
Key Concept: Use the tangent equation at parameter (a cos θ, b sin θ) and find where it intersects the coordinate axes extended. The key is recognizing that the distances CP and CQ can be expressed in terms of the parameter θ, then the algebraic sum simplifies due to trigonometric identity sin²θ + cos²θ = 1.
<p><strong>Step 1:</strong> Consider a point on the ellipse: (a cos θ, b sin θ). The tangent at this point is:</p><p>$$\frac{x \cos θ}{a} + \frac{y \sin θ}{b} = 1$$</p><p><strong>Step 2:</strong> Find intersection P with x-axis (y = 0):</p><p>$$\frac{x \cos θ}{a} = 1 \implies x = \frac{a}{\cos θ}$$</p><p>So CP² = $\frac{a^2}{\cos^2 θ}$ (distance from origin)</p><p><strong>Step 3:</strong> Find intersection Q with y-axis (x = 0):</p><p>$$\frac{y \sin θ}{b} = 1 \implies y = \frac{b}{\sin θ}$$</p><p>So CQ² = $\frac{b^2}{\sin^2 θ}$ (distance from origin)</p><p><strong>Step 4:</strong> Calculate the required expression:</p><p>$$\frac{a^2}{CP^2} + \frac{b^2}{CQ^2} = \frac{a^2}{\frac{a^2}{\cos^2 θ}} + \frac{b^2}{\frac{b^2}{\sin^2 θ}}$$</p><p>$$= \cos^2 θ + \sin^2 θ = 1$$</p><p>∴ <strong>Answer: 1</strong></p>
Correct Answer: 1