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 ________

<p>Given curve,</p> <p>$9{x^2} + 16{y^2} = 144$</p> <p>$\Rightarrow {{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$</p> <p>$\Rightarrow {{{x^2}} \over {{4^2}}} + {{{y^2}} \over {{3^2}}} = 1$</p> <p>$\therefore$ a = 4 and b = 3</p> <p>So, general point on the ellipse is $= (4\cos \theta ,3\sin \theta )$</p> <p>We know,</p> <p>Equation of tangent to a given ellipse at its point $(a\cos\theta ,b\sin \theta )$ is</p> <p>${{x\cos \theta } \over a} + {{y\sin \theta } \over b} = 1$</p> <p>$\therefore$ Here equation of tangent at point $(4\cos \theta ,3\sin \theta )$ is</p> <p>${{x\cos \theta } \over 4} + {{y\sin \theta } \over 3} = 1$</p> <p>When this tangent cut's x axis then y = 0.</p> <p>$\therefore$ ${{x\cos \theta } \over 4} + 0 = 1$</p> <p>$\Rightarrow x = 4\sec \theta$</p> <p>$\therefore$ Point of intersection at x axis is $A(4\sec \theta ,0)$.</p> <p>When this tangent cut's y axis then x = 0.</p> <p>$\therefore$ $0 + {{y\sin \theta } \over 3} = 1$</p> <p>$\Rightarrow y = 3\cos ec\theta$</p> <p>$\therefore$ Point of intersection at y axis is $B(0,3\cos ec\theta )$</p> <p>$\therefore$ Length of AB</p> <p>$= \sqrt {{{(4\sec \theta - 0)}^2} + {{(0 - 3\cos ec\theta )}^2}}$</p> <p>$= \sqrt {16{{\sec }^2}\theta + 9\cos e{c^2}\theta }$</p> <p>$= \sqrt {16(1 + {{\tan }^2}\theta ) + 9(1 + {{\cot }^2}\theta )}$</p> <p>$= \sqrt {25 + 16{{\tan }^2}\theta + 9{{\cot }^2}\theta }$</p> <p>We know, $AM \ge GM$</p> <p>$\therefore$ $${{16{{\tan }^2}\theta + 9{{\cot }^2}\theta } \over 2} \ge \sqrt {(16{{\tan }^2}\theta )(9{{\cot }^2}\theta )} $$</p> <p>$$ \Rightarrow 16{\tan ^2}\theta + 9{\cot ^2}\theta \ge 2(4\tan \theta )(3\cot \theta )$$</p> <p>$\Rightarrow 16{\tan ^2}\theta + 9{\cot ^2}\theta \ge 2 \times 4 \times 3$</p> <p>$\Rightarrow 16{\tan ^2}\theta + 9{\cot ^2}\theta \ge 24$</p> <p>$\therefore$ $AB = \sqrt {25 + 16{{\tan }^2}\theta + 9{{\cot }^2}\theta }$</p> <p>$\ge \sqrt {25 + 24}$</p> <p>$\ge \sqrt {49}$</p> <p>$\ge 7$</p> <p>$\therefore$ Minimum length of $AB = 7$.</p>