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

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-2

– 1/2

– 1/4

– 4

answer is A.

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Detailed Solution

Equation of the line be $y = m x + c$

Since it passes through (1, 2), 2 $= m + c$

$P (0, c), Q (- c / m, 0)$ If A is the area of the triangle, then

$A = \frac{1}{2} | c | |\frac{c}{m}| = \frac{c^{2}}{2 m} = \frac{(2 - m)^{2}}{2 m} = \frac{m^{2} - 4 m + 4}{2 m}$

$= \frac{m}{2} - 2 + \frac{2}{m}$

$\frac{d A}{d m} = 0 \Rightarrow \frac{1}{2} - \frac{2}{m^{2}} = \Rightarrow m^{2} = 4 \Rightarrow m = \pm 2$

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