let L1 be a straight line through the origin and L2 be the straight line x+y=1.if the intercept made by the circle x²+y²-x+3y=0 on L1 and L2 are equal, then which of the following represents L1?one answer is x-y=0 that answer i have got but how to get second answer that is x-7y=0

It’s simple
Lets say L1 is y=mx
Then as chords are equal the perpendicular distance from the center of the circle should also be same
center is
(1/2,-3/2)
so we have (by equating the perpendicular distance from center to L1 and L2)

squaring and simplifying we get the quadratic as

whose roots are
m=1,-1/7
so possible values of L1 are
x=y and
x = -7y