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If a pair of perpendicular lines through the origin together with the straight line $2 x+3 y=6$ form an isosceles triangle, then the area of that triangle (in sq units) is
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Verified Answer
The correct answer is:
$\frac{36}{13}$
Since, $\triangle A O B$ is a right angled triangle, so
$$
\angle A=\angle B=\frac{\pi}{4}
$$
$$
\begin{aligned}
So, \quad & A B=2 A P=2 O P \\
Now, \quad & O P=\frac{|6|}{\sqrt{4+9}}=\frac{6}{\sqrt{13}}
\end{aligned}
$$
So, Area of $\triangle A O B=\frac{1}{2} \times A B \times O P=O P^2=\frac{36}{13}$
$$
\angle A=\angle B=\frac{\pi}{4}
$$
$$
\begin{aligned}
So, \quad & A B=2 A P=2 O P \\
Now, \quad & O P=\frac{|6|}{\sqrt{4+9}}=\frac{6}{\sqrt{13}}
\end{aligned}
$$
So, Area of $\triangle A O B=\frac{1}{2} \times A B \times O P=O P^2=\frac{36}{13}$
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