Question

A line passes through $(2,2)$ and cuts a triangle of area $9$ square units from the first quadrant. The sum of all possible values for the slope of such a line, is?

• A. $-2.5$
• B. $-2$
• C. $-1.5$
• D. $-1$

Answer 1

Answer: (A)

$-2.5$

let the slope of line $=p$ and it pass through (2,2)

$\Rightarrow$ The equation will be given as

$\begin{array}{c}(y-2)=p(x-2)\\ \Rightarrow y=px-2p+2\end{array}$

Also, area of triangle, $A=\frac{1}{2}\times x\times y$

from (1)

$x=\frac{2p-2}{p}$

and $y=-2p+2$

$\begin{array}{cc}\therefore A& =\frac{1}{2}\times x\times y\\ \Rightarrow 2A=& (-2D+2)\times \frac{(2p-2)}{P}\\ \Rightarrow & =\frac{-4p^2+8p-4}{P}\\ & \text{Given that}A=9\text{unit}\\ \therefore & 18p=-4p^2+8p-4\\ \Rightarrow & 4p^2+10p+4=0\end{array}$

$\begin{array}{c}\Rightarrow \text{Sum of the slopes is}-\frac{10}{4}=-2.5\\ \text{Hence,}\left(A\right)\text{has to be the correct option.}\end{array}$