If A(-1,1) and B(2,3) are two fixed points, find the locus of a point P so that the area of $Δ P A B$ =8sq.units.

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Solution

$Let P(h,k) be any point on the locus. T h e n, A r e a (P A B) = 8 s q u n i t s \Rightarrow \frac{1}{2} | x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) | = 8 \Rightarrow \frac{1}{2} | - 1 (3 - k) + 2 (k - 1) + h (1 - 3) | = 18 \Rightarrow \frac{1}{2} | - 3 + k + 2 k - 2 - 2 h | = 8 \Rightarrow \frac{1}{2} | - 2 h + 3 k - 5 | = 8 \Rightarrow | - 2 h + 3 k - 5 | = 16 \Rightarrow - 2 h + 3 k - 5 = \pm 16 \Rightarrow 2 h + 3 k - 5 = \pm 16 = 0 \Rightarrow 2 h - 3 k + 21 = 0 o r, 2 h - 3 k - 11 = 0 H e n c e, t h e l o c u s o f (h, k) i s 2 x - 3 y + 21 = 0 o r, 2 x - 3 y - 11 = 0$

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