Question

Let a tangent to the curve $9 x^2+16 y^2=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $AB$ is

Answer 1

Correct Answer: 7

The equation of the ellipse is: \[ \frac{x^2}{16} + \frac{y^2}{9} = 1. \] The equation of the tangent at point \( P(4\cos\theta, 3\sin\theta) \) is: \[ \frac{x \cos\theta}{4} + \frac{y \sin\theta}{3} = 1. \] The tangent intersects the \( x \)-axis at \( A(4\sec\theta, 0) \) and the \( y \)-axis at \( B(0, 3\csc\theta) \). The length of \( AB \) is: \[ AB = \sqrt{(4\sec\theta)^2 + (3\csc\theta)^2}. \] Simplify: \[ AB = \sqrt{16\sec^2\theta + 9\csc^2\theta}. \] Using the identity: \[ \sec^2\theta + \csc^2\theta \geq 2, \] we find: \[ AB \geq \sqrt{25 + 16\tan^2\theta + 9\cot^2\theta} \geq 7. \] The minimum length of \( AB \) is: \[ \boxed{7}. \]