Question

A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle triangle $OPQ$, where $O$ is the origin. If the area of the triangle $OPQ$ is least, the the slope of the line $PQ$ is

• A. $-1/4$
• B. $-4$
• C. $-2$
• D. $-1/2$

Answer 1

The correct option is C $-2$

Equation of line passing through $(1,2)$ with slope $m$ is $y-2=m(x-1)$
Area of $\Delta OPQ=\frac{(m-2{)}^2}{2|m|}$
$\Delta =\frac{m^2-4m+4}{2m}$
$\Delta$ is least if $\frac{2}{m}=\frac{m}{2}$
$\Rightarrow m^2=4$
$\therefore m=2$ or $-2.$