If four points are A(6,3),B(−3,5),C(4,−2) and P(x,y), then the ratio of the areas of △PBC and △ABC is:
x+y−27
x−y−27
x−y+22
x+y+22
A
x−y+22
B
x+y+22
C
x+y−27
D
x−y−27
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Solution
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Given: Coordinates of points A(x1,y1)=(6,3),B(x2,y2)=(−3,5),C(x3,y3)=(4,−2) and P(x,y).
We know that the area of:
△PBC=12[x(y2−y3)+x3(y−y2)+x2(y3−y)]
=12[x(5+2)+4(y−5)−3(−2−y)]=12[7x+7y−14]
Similarly, the area of
△ABC=12[x1(y2−y3)+x2(y3−y1)]+x3(y1−y2)
=12[6(5+2)−3(−2−3)+4(3−5)]=492
Therefore, the ratio of the areas of △PAB and △ABC
=7x+7y−1449=7(x+y−2)49=x+y−27
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