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  4. The point A divides the line segment joining the points (.-5,1.) and (.3,5.) in the ratio k:1 internally and the coordinates of points B and C are (1,5) and (7 , - 2) respectively. If the area of Δ ABC is 2 sq. units, then the sum of all the values of k is equal to

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Q. The point $A$ divides the line segment joining the points $\left(\right.-5,1\left.\right)$ and $\left(\right.3,5\left.\right)$ in the ratio $k:1$ internally and the coordinates of points $B$ and $C$ are $\left(1,5\right)$ and $\left(7 , - 2\right)$ respectively. If the area of $\Delta ABC$ is $2$ sq. units, then the sum of all the values of $k$ is equal to

NTA AbhyasNTA Abhyas 2020Straight Lines

Solution:

$A\equiv \left(\frac{3 k - 5}{k + 1} , \frac{5 k + 1}{k + 1}\right)$
Area of $\Delta ABC=2 sq.units$
$\Rightarrow \frac{1}{2}\left[\frac{3 k - 5}{k + 1} \left(5 + 2\right) + 1 \left(- 2 - \frac{5 k + 1}{k + 1}\right) + 7 \left(\frac{5 k + 1}{k + 1} - 5\right)\right]=\pm2$
$\Rightarrow 14k-66=\pm4\left(k + 1\right)\Rightarrow k=7$ or $\frac{31}{9}$