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 $△ABC$ where $B$ is $(1,5)$ and $C$ is $(7,-2)$ is equal to $2$ units in magnitude is _______

Answer 1

Answer: 7

Using section formula $A\left(\frac{3k-5}{k+1},\frac{5k+1}{k+1}\right)$
Area of triangle $ABC$ is $2sq$. units
$\Rightarrow \frac{1}{2}\left|\begin{array}{ccc}1& 5& 1\\ 7& -2& 1\\ \frac{3k-5}{k+1}& \frac{5k+1}{k+1}& 1\end{array}\right|=\pm 2$
Operating $R_2\to R_2-R_1;R_3\to R_3-R_1$
$\left|\begin{array}{ccc}1& 5& 1\\ 6& -7& 0\\ \frac{3k-5}{k+1}-1& \frac{5k+1}{k+1}-5& 0\end{array}\right|=\pm 4$
$\Rightarrow 6\left(\frac{5k+1-5k-5}{k+1}\right)+7\left(\frac{3k-5-k-1}{k+1}\right)=\pm 4$
$\Rightarrow -24+7(2k-6)=\pm 4(k+1)$
$\Rightarrow k=7$ or $k=\frac{31}{9}$