Problem 67 Hard Difficulty

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 $.

Answer

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Video Transcript

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