find the centroid and area of the triangle formed by the lines 2y²-xy-6x² =0,x+y+4=0
Answers
we have to find the centroid and area of the triangle formed by the lines 2y² - xy - 6x² = 0 and x + y + 4 = 0.
solution : here 2y² - xy - 6x² = 0
⇒2y² - 4xy + 3xy - 6x² = 0
⇒2y(y - 2x) + 3x(y - 2x) = 0
⇒(2y + 3x)(y - 2x) = 0
⇒3x + 2y = 0 ....(1)
and -2x + y = 0 ....(2)
given, x + y + 4 = 0 ....(3)
from equations (1) and (2), x = 0, y = 0
from equations (1) and (3), x = 8, y = -12
from equations (2) and (3), x = -4/3, y = -8/3
centroid = [ (x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3]
= [(0 + 8 - 4/3)/3 , (0 - 12 - 8/3)/3]
= [20/9, -44/9]
area of triangle = 1/2 [0(-12 + 8/3) + 8(-8/3 - 0) + (-4/3)(0 + 12)]
= 1/2 [ 0 - 64/3 - 48/3]
= 1/2 × 112/3
= 56/3 sq unit
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