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| Manipulate[ Block[{a = 1, b = 2}, h = (a Cos[t0] b Sin[t0]^3 + b a Cos[t0]^3 Sin[t0])/(b Sin[t0]); k = -((2/(3 a Cos[t0]))/(2/(3 b Sin[t0]))) (-a Cos[t0]^3) + b Sin[t0]^3; Show[ {RegionPlot[(x/a)^(2/3) + (y/b)^(2/3) < 1, {x, 0, a}, {y, 0, b}], ParametricPlot[{(a Cos[t] b Sin[t]^3 + b a Cos[t]^3 Sin[t])/( b Sin[t]), -((2/(3 a Cos[t]))/(2/(3 b Sin[t]))) (-a Cos[t]^3) + b Sin[t]^3}, {t, 0, Pi}], Plot[{-((2/(3 a Cos[t0]))/(2/(3 b Sin[t0]))) (x - a Cos[t0]^3) + b Sin[t0]^3}, {x, 0, ( a Cos[t0] b Sin[t0]^3 + b a Cos[t0]^3 Sin[t0])/(b Sin[t0])}, PlotStyle -> Red], Graphics[{Dotted, Blue, Line[{{0, k}, {h, k}, {h, 0}}], Line[{{h, k}, {a Cos[t0]^3, b Sin[t0]^3}}], PointSize[0.02], Point[{{0, k}, {h, 0}, {h, k}, {a Cos[t0]^3, b Sin[t0]^3}}]}] } ] ], {t0, 0.3, Pi/2 - 0.3} ]
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