A triangle ABC is inscribed in the ellipse 2x2 + y2 = 18. If B is (1,4) and C is (3,0), then the greatest value of the area of the triangle
ABC is (take 6 = 2.45) A.Value of [4] [-]represents greatest integer function.
Answers
Given : A triangle ABC is inscribed in the ellipse 2x²+ y2 = 18. If B is (1,4) and C is (3,0),
To Find : the greatest value of the area of the triangle
(A) 3√6- 6 (B) √6+2 (0)3√6+6 (D) 6/2+4
Solution:
B ( 1 , 4)
C ( 3, 0)
A = ( x , y)
2x² + y² = 18
A = ( x , y) , B ( 1 , 4) , C ( 3, 0)
Area of Triangle
Ar = (1/2) | x ( 4 - 0) + 1 ( 0 - y) + 3(y - 4) |
=> Ar = (1/2) | 4x - y + 3y - 12 |
=> Ar = (1/2) | 4x + 2y - 12 |
=> Ar = | 2x + y - 6 |
2x + y - 6 > 0
Ar = 2x + y - 6
=> Ar = 2x + √(18 - 2x²)
=> d(Ar)/dx = 2 + (1/2√(18 - 2x²) )(-4x)
=> d(Ar)/dx = 0
=> 2 + (1/2√(18 - 2x²) )(-4x) = 0
=> 2 = 2x/√(18 - 2x²)
=> √(18 - 2x²) = x
=> 18 - 2x² = x²
=> x² = 6 => y² = 6
=> x = ±√6 , y = ±√6
2x + y - 6 > 0 => x & y are +√6
Area = 2x + y - 6
= 3√6 - 6
2x + y - 6 < 0 Then x & y are - √6 or x = -√6 , y = √6 or x = √6 , y = -√6
x & y are - √6
Area = | -3√6 - 6|
= 3√6 + 6
x = -√6 , y = √6 = | -√6 - 6| = √6 + 6
x = √6 , y = -√6 = | √6 - 6 | = 6 - √6
3√6 + 6 is the maximum area
= 3(2.45) + 6
= 13.35 sq unit
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