Question-66211

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

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

$${Find}\:{the}\:{maximum}\:{area}\:{of}\:{a}\:{right} \\ $$$${triangle}\:{inscribed}\:{in}\:{an}\:{ellipse}. \\ $$

Commented by MJS last updated on 10/Aug/19

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

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

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

$${thank}\:{you}\:{sir}! \\ $$$${can}\:{we}\:{say}\:{that}\:{the}\:{maximum}\:{triangle} \\ $$$${is}\:{always}\:{isosceles}\:{with}\:{C}\:{on}\:{the} \\ $$$${major}\:{or}\:{minor}\:{axis}? \\ $$

Commented by MJS last updated on 11/Aug/19

draw a circle with radius a inscribe an equilateral triangle with C= ((0),(a) ) rotate it by any angle θ transform the whole thing: P= ((x),(y) ) → P′= ((x),(((by)/a)) ) all possible triangles have the same area which is the max. area of triangles inscribed in the ellipse (x^2 /a^2 )+(y^2 /b^2 )=1

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

Answered by MJS last updated on 11/Aug/19

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

$${thanks}\:{again}\:{sir}! \\ $$

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