A) \[\frac{20}{3}\,\text{sq}\,\text{units}\]
B) \[3\,\text{sq}\,\text{units}\]
C) \[\frac{10}{3}\,\text{sq}\,\text{units}\]
D) \[\text{2}\,\text{sq}\,\text{units}\]
Correct Answer: C
Solution :
The equations of given lines are \[x=2\] and \[4{{x}^{2}}-9xy-9{{y}^{2}}=0\] \[\Rightarrow \]\[4x(x-3y)+3y(x-3y)=0\] \[\Rightarrow \]\[(x-3y)(4x+3y)=0\] \[\Rightarrow \]\[(x-3y)=0\]and \[(4x+3y)=0\] So, we have the three sides of triangle are \[x=2,\,x-3y=0\]and \[4x+3y=0\] Solving these equations taking any two at a time, we get the vertices of the triangle \[(0,0),(2,-8/3)\]and\[(2,2/3).\] Now, area of the triangle \[=\frac{1}{2}\left| \begin{matrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & -8/3 & 2/3 \\ \end{matrix} \right|\] \[=\frac{1}{2}\left[ 2.\frac{2}{3}+2.\frac{8}{3} \right]\] \[=\frac{1}{2}.\frac{20}{3}=\frac{10}{3}\,\text{sq}\,\text{units}\]
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