Let the equation of the pair of lines, y = px and y = qx, can be written as (y - px) (y - qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 - 4xy - 5y2 = 0 is :
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Official solution
Equation of angle bisector of homogeneous
equation of pair of straight line ax2
+ 2hxy + by2
is
\[{{{x^2} - {y^2}} \over {a - b}} = {{xy} \over h}\]
for x2 -- 4xy -- 5y2
= 0
a = 1, h = -- 2, b = -- 5
So, equation of angle bisector is
\[{{{x^2} - {y^2}} \over {1 - ( - 5)}} = {{xy} \over { - 2}}\]
\[{{{x^2} - {y^2}} \over 6} = {{xy} \over { - 2}}\]
\[ \Rightarrow {x^2} - {y^2} = - 3xy\]
So, combined equation of angle bisector is x2+3xy−y2=0