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Question

The equation 9x3+9x2y45x2=4y3+4xy220y2 represents 3 straight lines, two of which passes through origin. Then find the area of the triangle formed by these lines.

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Solution

9x3+9x2y45x2=4y3+4xy220y2

9x2(x+y5)=4y2(x+y5)

9x2(x+y5)4y2(x+y5)=0

(9x24x2)(x+y5)=0

(3x+2y)(3x2x)(x+y5)=0

so the three lines are

x=23y, x=23y and x+y5

from equation of lines , we get the vertices of triangle formed by the lines

A(0,0),B(10,15) and C(2,3)

Area of the triangle =12(xaxc)(ybya)(xaxb)(ycya)

=12(xayb+xbyc+xcyaxaycxcybxbya)

=12(0×15+10×3+2×00×32×15(10×0)
=12×60

since, area is non- negativengle

So, the area of the triangle=30unit2

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