I have taken a parameter $(a \cos c, b \sin c)$ where $c$ is the eccentric angle and the tangent passing through this point cuts the x-axis at the point $(a \cos c, 0)$ and y-axis at $(0,b \sin c)$.
After this I have calculated the the length using Pythagoras theorem. But I couldn't get the minimum value.
Length of tangent intercepted: $\sqrt { \frac {a^2}{ {cos}^2 t}+ \frac {b^2 }{{sin}^2t}}$ = $\sqrt { a^2 {sec}^2 t+ b^2 {cosec}^2t}$ = $\sqrt { a^2{tan}^2t+a^2+ b^2 {cot}^2t+b^2}$ Using AM-GM inequality: $\frac {a^2{tan}^2t+b^2 {cot}^2t} 2 \ge \sqrt {a^2{tan}^2t.b^2 {cot}^2t}$ So ${a^2{tan}^2t+b^2 {cot}^2t} \ge 2ab $ Therefore required minimum value is $a+b$
