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Question Number 66211 by mr W last updated on 10/Aug/19

Commented by mr W last updated on 10/Aug/19

$$Findthemaximumareaofaright$$$$triangleinscribedinanellipse.$$

Commented by MJS last updated on 10/Aug/19

$$\mathrm{I}\mathrm{have}\mathrm{got}\mathrm{no}\mathrm{time}\mathrm{right}\mathrm{now},\mathrm{but}\mathrm{the}\mathrm{centroid}$$$$\mathrm{of}\mathrm{the}\mathrm{triangle}\mathrm{must}\mathrm{be}\mathrm{the}\mathrm{center}\mathrm{of}\mathrm{the}$$$$\mathrm{ellipse}.$$$$\mathrm{I}\mathrm{once}\mathrm{posted}\mathrm{the}\mathrm{equations}\mathrm{of}\mathrm{the}\mathrm{ellipse}$$$$\mathrm{with}\mathrm{minimal}\mathrm{area}\mathrm{surrounding}\mathrm{any}\mathrm{triangle}$$$$\mathrm{it}\mathrm{should}\mathrm{be}\mathrm{possible}\mathrm{to}\mathrm{reverse}\mathrm{this}\mathrm{system}$$$$\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{family}\mathrm{of}\mathrm{triangles}\mathrm{with}\mathrm{maximal}$$$$\mathrm{area}\mathrm{inscribed}\mathrm{in}\mathrm{any}\mathrm{ellipse}(=\mathrm{all}\mathrm{triangles}$$$$\mathrm{with}\mathrm{vertices}\mathrm{on}\mathrm{the}\mathrm{ellipse}\mathrm{and}\mathrm{centroid}\mathrm{in}$$$$\mathrm{the}\mathrm{center}\mathrm{of}\mathrm{the}\mathrm{ellipse})\mathrm{and}\mathrm{to}\mathrm{find}\mathrm{the}$$$$\mathrm{right}\mathrm{angled}\mathrm{one}\left(\mathrm{s}\right)\mathrm{out}\mathrm{of}\mathrm{this}\mathrm{family}$$

Commented by mr W last updated on 11/Aug/19

$$thankyousir!$$$$canwesaythatthemaximumtriangle$$$$isalwaysisosceleswithConthe$$$$majororminoraxis?$$

Commented by MJS last updated on 11/Aug/19

$$\mathrm{draw}\mathrm{a}\mathrm{circle}\mathrm{with}\mathrm{radius}a$$$$\mathrm{inscribe}\mathrm{an}\mathrm{equilateral}\mathrm{triangle}\mathrm{with}C=\left(\begin{array}{c}0\\ a\end{array}\right)$$$$\mathrm{rotate}\mathrm{it}\mathrm{by}\mathrm{any}\mathrm{angle}\theta$$$$\mathrm{transform}\mathrm{the}\mathrm{whole}\mathrm{thing}:P=\left(\begin{array}{c}x\\ y\end{array}\right)\to P^{\prime }=\left(\begin{array}{c}x\\ \frac{by}{a}\end{array}\right)$$$$\mathrm{all}\mathrm{possible}\mathrm{triangles}\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{area}$$$$\mathrm{which}\mathrm{is}\mathrm{the}\max .\mathrm{area}\mathrm{of}\mathrm{triangles}\mathrm{inscribed}$$$$\mathrm{in}\mathrm{the}\mathrm{ellipse}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

Answered by MJS last updated on 11/Aug/19

Commented by mr W last updated on 12/Aug/19

$$thanksagainsir!$$

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