let P be a variable point on the ellipse x2/a2 + y2/b2 = 1 with R and S as the - Maths - Conic Sections

Question

let P be a variable point on the ellipse
x2/a2 + y2/b2 = 1 with
R and S as the two focii Find the maximum area of the triangle PRS

Rishabh Mittal · Accepted Answer

Equation of ellipse is x2a2+y2b2=1fociiae,0From equation of ellipsey2b2=1-x2a2⇒y=baa2-x2
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vertces of triangle PRS Px,y,R-ae,0 and Sae,0so, length of base=distance between R and S =2ae
Height=y
So, area of triangle,A=12×base×height=12×2ae×y              =12×2ae×baa2-x2     sincey=baa2-x2              =eba2-x2Differntiatin both sides with respect to xdAdx=eb12a2-x2-2x=-ebxa2-x2put dAdx to find critical point
-ebxa2-x2=0⇒x=0
now,d2Adx2=-eba2-x2-x12a2-x2-2xa2-x22At x=0 d2Adx2=-eba-0a2=-ebaas d2Adx2 is negative at x=0 so, it is a point of maxima
At x=0y=baa2-0=bSo, maximum area of triangle,A=12×2ae×b=aeb     =aba2-b2a=ba2-b2          since e=a2-b2a

let P be a variable point on the ellipse x2/a2 + y2/b2 = 1 with R and S as the two focii. Find the maximum area of the triangle PRS

let P be a variable point on the ellipse x²/a² + y²/b²= 1 with R and S as the two focii. Find the maximum area of the triangle PRS