A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a…
A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $OPQ$, where $O$ is the origin. If the area of the triangle $OPQ$ is least, then the slope of the line $PQ$ is
$-\frac{1}{4}$
$-4$
$-2$
$-\frac{1}{2}$
Solution
Equation of line passing through $(1,2)$ with slope $m$ is $y-2=m(x-1)$
Area of $\Delta \mathrm{OPQ}=\frac{(m-2)^2}{2|m|}$
$\Delta=\frac{m^2+4-4 m}{2 m} \quad \Delta=\frac{m}{2}+\frac{2}{m}-2$
$\Delta$ is least if $\frac{m}{2}=\frac{2}{m} \quad \Rightarrow m^2=4 \quad \Rightarrow m=\pm 2 \quad \Rightarrow m=-2$