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Question

Let L1 be a straight line passing through the origin and L2 be the straight line x+y=1. If the intercepts made by the circle x2+y2−x+3y=0 on L1 and L2 are equal, then which of the following equations can represent L1

A
x+y=0
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B
xy=0
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C
x+7y=0
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D
x7y=0
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Solution

The correct options are
B xy=0
D x+7y=0
Let the equation of the line passing through origin be y=mx. Therefore
x2+y2x+3y=0x2+m2x2x+3mx=0

x[x(1+m2)(13m)]=0

x=0,x=13m1+m2 and

y=0,y=m13m1+m2

Intercept =(13m1+m2)2+m2(13m1+m2)2=13m1+m21+m2=13m1+m2

Similarly for the line x+y=1y=1x

x2+(1x)2x+3(1x)=0x2+x22x+1x+33x=0

x=1,2 and y=0,1

Intercept (12)2+(0+1)2=2

Given that 2+2m2=9m26m+17m26m1=0

(7m+1)(m1)=0m=1,17

Lines are y=x,y=17x or xy=0,x+7y=0

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