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Question

The equations xy=4 and x2+4xy+y2=0 represent the sides of

A
an equilateral triangle
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B
a right angled triangle
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C
an isosceles triangle
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D
a right angled isosceles triangle
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Solution

The correct option is A an equilateral triangle
x2+4xy+y2=0(yx)2+4yx+1=0yx=2±3
Acute angle between the lines is tan1(241)1+1=π3
Now angle made by the line xy=4 and yx=2±3
=tan1∣ ∣2±311+(2±3)∣ ∣=tan1∣ ∣3(3±1)1±3∣ ∣=π3
Hence the lines will form a equilateral triangle

Alternate Solution:
Acute angle between pair of lines x2+4xy+y2=0
=tan122212=π3
the line y=x+c makes equal angle with x2+4xy+y2=0
[y=x doesn't change the pair of lines x2+4xy+y2=0]
all angles of the triangle are equal to 60°
Hence triangle is equilateral triangle

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