Find the area of the greatest isosceles triangle that can be inscribed in the ellipse (a2x2)+(b2y2)=1
having its vertex coincident with one extremity of the major axis.

Question

Find the area of the greatest isosceles triangle  that can be inscribed in the ellipse (a2x2​)+(b2y2​)=1
having its vertex coincident with one extremity of the  major axis.

Filo · Accepted Answer

Solution: construct the traingle as shown in figure and name it PAQ
in

∠ A P Q = \frac{1}{2} P Q A D

= \frac{1}{2} \cdot 2 (P D) (O A - O D)

In

∠ A P Q \Rightarrow (B sin θ) (a - a cos θ)

= a b sin θ - a b sin θ cos θ

= \frac{a b}{2} \cdot [2 sin θ - 2 sin θ cos θ]

= \frac{a b}{2} \cdot [2 sin θ - sin 2 θ]

as,

\frac{d ∠}{d θ} = 0

= (\frac{a b}{2}) [2 cos θ - 2 cos 2 θ] = 0

= 2 cos θ - 2 cos 2 θ = 0

= cos θ - cos 2 θ = 0

= cos θ - (2 cos^{2} θ - 1) = 0

= 2 cos^{2} θ - 2 cos θ + cos θ - 1 = 0

= 2 cos θ (cos θ - 1) + 1 (cos θ - 1) = 0

= (2 cos θ + 1) (cos θ - 1) = 0

= cos θ = - \frac{1}{2}

a t θ = \frac{2 π}{3}

area =

a b [\frac{3}{2} + \frac{3}{2} \cdot \frac{1}{2}

= a b [\frac{3}{2} + \frac{3}{4}]

= 3 \frac{3}{4} a b

answer

Question Text	Find the area of the greatest isosceles triangle that can be inscribed in the ellipse $(\frac{x ^{2}}{a ^{2}}) + (\frac{y ^{2}}{b ^{2}}) = 1$ having its vertex coincident with one extremity of the major axis.
Answer Type	Text solution:1
Upvotes	150

Find the area of the greatest isosceles triangle that can be inscribed in the ellipse $(\frac{x ^{2}}{a ^{2}}) + (\frac{y ^{2}}{b ^{2}}) = 1$ having its vertex coincident with one extremity of the major axis.

Text solutionVerified

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Find the area of the greatest isosceles triangle that can be inscribed in the ellipse (a2x2​)+(b2y2​)=1 having its vertex coincident with one extremity of the major axis.

Text solutionVerified

Practice questions from similar books

Find the area of the greatest isosceles triangle that can be inscribed in the ellipse $(\frac{x ^{2}}{a ^{2}}) + (\frac{y ^{2}}{b ^{2}}) = 1$ having its vertex coincident with one extremity of the major axis.