Class 12 Applications Of Derivatives · Very Long Answer5 Marksgeometric_optimization

Class 12 Maths · Applications Of Derivatives · Very Long Answer #82

For the ellipse x²/a²+y²/b²=1 with a≥b>0, find the maximum area of an inscribed isosceles triangle whose apex is (a,0), with its base perpendicular to the major axis.

Isosceles triangle with apex on major axis

✅ Step-by-Step Solution

Final Answer: 3√3ab/4

Take base vertices (a cosθ,±b sinθ), 0≤θ≤π, and apex (a,0).
The area is A=ab sinθ(1−cosθ).
A′=ab[1+cosθ−2cos²θ]=ab(1−cosθ)(2cosθ+1), giving the interior critical point θ=2π/3.
For 0<θ<2π/3 the derivative is positive and for 2π/3<θ<π it is negative; endpoint areas vanish.
Amax=ab(√3/2)(3/2)=3√3ab/4.

📋 CBSE Stepwise Marking Scheme

[1 mark] Take base vertices (a cosθ,±b sinθ), 0≤θ≤π, and apex (a,0).
[1 mark] The area is A=ab sinθ(1−cosθ).
[1 mark] A′=ab[1+cosθ−2cos²θ]=ab(1−cosθ)(2cosθ+1), giving the interior critical point θ=2π/3.
[1 mark] For 0<θ<2π/3 the derivative is positive and for 2π/3<θ<π it is negative; endpoint areas vanish.
[1 mark] Amax=ab(√3/2)(3/2)=3√3ab/4.

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Class 12 Maths · Applications Of Derivatives · Very Long Answer #82

✅ Step-by-Step Solution

📋 CBSE Stepwise Marking Scheme

Master Chapter 6: Applications Of Derivatives

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