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 $OPQ$, where $O$ is the origin. If the area of the triangle $OPQ$ is least, then the slope of the line $PQ$ is

• A. - $\frac{1}{4}$
• B. $-4$
• C. $-2$
• D. - $\frac{1}{2}$

Answer 1

The correct option is C $-2$

Line equation will be $(y-2)=m(x-1)$
Put $y=0$ we will get $x$ intercept $=1-\frac{2}{m}$
Put $x=0$ we will get $y$ intercept $=2-m$
Area of triangle $=\frac{1}{2}$ ($x$ intercept)($y$ intercept) $=\frac{1}{2}(1-\frac{2}{m})(2-m)$
$=\frac{1}{2}(2-m-\frac{4}{m}+2)$
$=\frac{1}{2}(4-(m+\frac{4}{m}))$
It will be least when value of $m$ will be $-2.$