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Question

Show that the lines x2+4xy+y2=0 and xy=4 form an equilateral triangle. Also find the area of the triangle.

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Solution

Given pair of lines are x2+4xy+y2=0

(yx)2+4yx+1=0

yx=4±232=2±3,

The lines y=(2+3)x and y=(23)x and xy=4 forms an equilateral triangle

Clearly the pair of lines x2+4xy+y2=0 intersect at origin,

The perpendicular distance form origin to xy=4 is the height of the equilateral triangle

h=22

Area of the triangle is h23=83

847305_855380_ans_5a4afbaf0e48476e9bb651f6189759b3.jpg

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