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Question

Prove that the lines:
(a+b)x+(ab)y2ab=0.......(1)
(ab)x+(a+b)y2ab=0.......(2)
and x+y=0......(3) form an isosceles triangle whose vertical angle is 2tan1(a/b).

Solution
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Slopes of the given lines are respectively
a+bab,aba+b,1
The triangle will be isosceles if x+y=0 is equally inclined to other two whose slopes are written above
1+(a+bab)1+(a+bab)=1+(a+bab)1+aba+b=tanθ
tanθ=b/a
is isosceles whose vertical angle is
π2θ=2(π2tan1ba)=2(cot1ba)=2tan1ab
1029290_1007282_ans_01e9cd15b8d447838fc88db2630cad52.PNG

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