Write the value of θ ϵ $(0, \frac{π}{2})$ for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.

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Solution

Let A be the area of the triangle formed by the points O (0,0), A (acosθ,bsinθ) and B (acosθ,− bsinθ)

$A = \frac{1}{2} |\begin{matrix} 0 & 0 & 1 \\ a cosθ & b sinθ & 1 \\ a cosθ & - b sinθ & 1 \end{matrix}| \Rightarrow A = \frac{1}{2} |(- a b sinθcosθ - a b sinθcosθ)| \Rightarrow A = a b sinθcosθ = \frac{1}{2} \sin 2 θ$
Now,
$∴ A_{m a x} = \frac{1}{2}, when \sin 2 θ = 1 \Rightarrow ∴ A_{m a x} = \frac{1}{2}, when 2 θ = \frac{π}{2} \Rightarrow θ = \frac{π}{4}$

Hence, the area of the triangle formed by the given points is maximum when $θ = \frac{π}{4}$ .

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Write the value of θ ϵ 0,π2 for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.

Write the value of θ ϵ $(0, \frac{π}{2})$ for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.