If $A (6, 3), B (3, 5), C (4, 2), P (α, β)$ then the ratio of the area of the triangles $P B C, A B C$ is

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$| α - β | : 7$

$| α + β + 2 | : 7$

$| 3 α + β - 14 | : 7$

$| α + β | : 7$

answer is D.

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Detailed Solution

$[P B C] = \frac{1}{2} |\begin{matrix} α & β & 1 \\ 3 & 5 & 1 \\ 4 & 2 & 1 \end{matrix}|$

$= \frac{1}{2} | α (3) - β (- 1) + 1 (- 14) |$

$= \frac{1}{2} | 3 α + β - 14 |$

$[A B C] = \frac{1}{2} |\begin{matrix} 6 & 3 & 1 \\ 3 & 5 & 1 \\ 4 & 2 & 1 \end{matrix}|$

$= \frac{1}{2} | 18 - 3 (- 1) + 1 (- 14) |$

$= \frac{7}{2}$

$∴ [P B C] : [A B C] = | 3 α + β - 14 | : 7$

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