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The point A divides the join of $P (- 5, 1)$ and $Q (3, 5)$ in the ratio $k : 1$ . The values of k for which the area of $Δ A B C$ where $B (1, 5), C (7, - 2)$ is 2 sq.units is

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$- 7, - 31 / 9$

$7, - 31 / 9$

$7, 31 / 9$

$- 7, 31 / 9$

answer is A.

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Detailed Solution

$A = (\frac{3 k - 5}{k + 1}, \frac{5 k + 1}{k + 1})$ Area of triangle ABC = 2

$\Rightarrow \frac{1}{2} |\begin{matrix} 1 - 7 & 5 + 2 \\ 1 - \frac{3 k - 5}{k + 1} & 5 - \frac{5 k + 1}{k + 1} \end{matrix}| = 2$ $\Rightarrow |\begin{matrix} - 6 & 7 \\ \frac{- (2 k - 6)}{k + 1} & \frac{4}{k + 1} \end{matrix}| = 4$

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