Question

The point $A$ divides the line segment joining $P(-5,1)$ and $Q(3,5)$ in the ratio $k: 1$. The integral value of $k$ for which the area of $\triangle A B C$ where $B$ is $(1,5)$ and $C$ is $(7,-2)$ is equal to $2$ units in magnitude is _______

Answer 1

Using section formula $A\left(\frac{3 k-5}{k+1}, \frac{5 k+1}{k+1}\right)$
Area of triangle $A B C$ is $2 sq$. units
$\Rightarrow \frac{1}{2}\begin{vmatrix} 1 & 5 & 1 \\ 7 & -2 & 1 \\ \frac{3 k-5}{k+1} & \frac{5 k+1}{k+1} & 1 \end{vmatrix}=\pm 2$
Operating $R_{2} \rightarrow R_{2}-R_{1} ; R_{3} \rightarrow R_{3}-R_{1}$
$\begin{vmatrix}1 & 5 & 1 \\6 & -7 & 0 \\ \frac{3 k-5}{k+1}-1 & \frac{5 k+1}{k+1}-5 & 0 \end{vmatrix}=\pm 4 $
$\Rightarrow 6\left(\frac{5 k+1-5 k-5}{k+1}\right)+7\left(\frac{3 k-5-k-1}{k+1}\right)=\pm 4$
$\Rightarrow -24+7(2 k-6)=\pm 4(k+1)$
$\Rightarrow k=7$ or $k=\frac{31}{9}$