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  4. A straight line l1 with equation x-2y+10=0 meets the circle with equation x2+y2=100 at B in the first quadrant. A line through B, perpendicular to l1 cuts the x -axis and y -axis at P and Q respectively. The area (in sq. units) of the triangle OPQ is (where, O is the origin)

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Q. A straight line $l_{1}$ with equation $x-2y+10=0$ meets the circle with equation $x^{2}+y^{2}=100$ at $B$ in the first quadrant. A line through $B,$ perpendicular to $l_{1}$ cuts the $x$ -axis and $y$ -axis at $P$ and $Q$ respectively. The area (in sq. units) of the triangle $OPQ$ is (where, $O$ is the origin)

NTA AbhyasNTA Abhyas 2020Conic Sections

Solution:

Solution
Slope of $l_{1}=\frac{1}{2}$
Slope of $l_{2}=-2$
Equation of $l_{2}$ is $y=-2\left(x - 10\right)$
$\Rightarrow y+2x=20$
Hence, $P$ and $Q$ are $\left(10 , 0\right)$ and $\left(0 , 20\right)$
Area of $\Delta OPQ=\frac{1}{2}\times 10\times 20=100$ sq. units