Write the value of $θ ε (\frac{π}{2})$ for which area of the triangle formed by points O(0,0), A $(a c o s θ, b s i n θ)$ and $(a c o s θ, b s i n θ)$ is maximam.

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Solution

Area of triangle formed by O(0,0), A $(a c o s θ, b s i n θ)$ and $(a c o s θ, b s i n θ)$

$A = \frac{1}{2} [\begin{matrix} x_{1} (u_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) \end{matrix}]$

$= [\begin{matrix} \frac{1}{2} [\begin{matrix} 0 (b s i n θ) + a c o s θ (- b s i n θ - 0) + a c o s θ (0 - b s i n θ) \end{matrix}] \end{matrix}]$

$= [\begin{matrix} \frac{1}{2} [- a b c o s θ s i n θ - a b s i n θ c o s θ] \end{matrix}]$

$= [\begin{matrix} \frac{1}{2} [- 2 a b s i n θ c o s θ] \end{matrix}]$

$= | - a b s i n θ c o s θ |$

$= \frac{a b}{2} | s i n 2 θ |$

A is maximum if $s i n 2 θ = 1$

$2 θ = \frac{π}{2}$

$θ = \frac{π}{4}$

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