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Consider the tangent line to the ellipse $ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 $ at a point $ (p, q) $ in the first quadrant.

(a) Show that the tangent line has $ x $-intercept $ a^2/p $ and $ y $-intercept $ b^2/q $.

(b) Show that the portion of the tangent line cut off by the coordinates axes has minimum length $ a + b $.

(c) Show that the triangle formed by the tangent line and the coordinate axes has minimum area $ ab $.

a)

$\begin{aligned} X-i n t e r c e p t &=\frac{a^{2}}{p} \\ Y-i n t e r c e p t &=\frac{b^{2}}{q} \end{aligned}$

b)

Minimum length $=a+b$

c)

Minimum Area formed is ab!

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