Question

find the maximum area of isosceles triangle inscribed in the eclipse having lunch of major axis 10 units and minor axis 8 units​

Answer 1

Step-by-step explanation:

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Answer 2

Answer

Solution:

Consider the isosceles triangle ABC:

A(a,0),B(acosθ,bsinθ) and C(acosθ,−bsinθ)

Area of △ABC=  

2

1

​

×BC× height of △ABC

Height of △ABC=a(1+cosθ)

BC=2bsinθ

or, △=  

2

1

​

×2bsinθ×a(1+cosθ)=absinθ(1+cosθ)

For maximum area of the triangle,

dθ

d△

​

=abcosθ(1+cosθ)−absin  

2

θ=0

or, cosθ(1+cosθ)−sin  

2

θ=0

or, cosθ(1+cosθ)−(1+cosθ)(1−cosθ)=0

or, (1+cosθ)(cosθ−1+cosθ)=0

or, (1+cosθ)(2cosθ−1)=0

or, cosθ=−1 or, cosθ=  

2

1

​

 

For maximum value, we take  

cosθ=  

2

1

​

 and sinθ=  

2

3

​

 

​

 

or, Maximum area =ab×  

2

3

​

 

​

(1+  

2

1

​

)=  

4

3  

3

​

 

​

ab