$The minimum area of the triangle formed by the \tan gent to the ellipse$ $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ 

$and the coordinate axes is$

$\text{ Equation of the tangent at }P\left(\theta \right)\text{ to the given ellipse is }\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$

$\text{ ∴Area of the triangle formed by it with coordinate axis =}\frac{ab}{2\sin \theta \cos \theta }\text{=}\frac{ab}{\sin \text{2θ}}\ge ab$

$\therefore minimun area=ab$