Area of triangle formed by x²-4xy+y²=0, x+y=3 is .....
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Show that the lines x
2
+4xy+y
2
=0 and x−y=4 form an equilateral triangle. Also find the area of the triangle.
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ANSWER
Given pair of lines are x
2
+4xy+y
2
=0
⟹(
x
y
)
2
+4
x
y
+1=0
⟹
x
y
=
2
−4±2
3
=−2±
3
,
∴ The lines y=(−2+
3
)x and y=(−2−
3
)x and x−y=4 forms an equilateral triangle
Clearly the pair of lines x
2
+4xy+y
2
=0 intersect at origin,
The perpendicular distance form origin to x−y=4 is the height of the equilateral triangle
h=2
2
∴ Area of the triangle is
3
h
2
=
3
8
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