Let `L_1` be a straight line passing through the origin and ` L_2` be the straight line `x + y = 1` if the intercepts made by the circle `x^2 + y^2-x+ 3y = 0` on `L_1` and `L_2` are equal, then which of the following equations can represent `L_1`?

Options

A. x+ y = 0
B. x - y = 0
C. x + 7y = 0
D. x - 7y = 0

Answer

B, C

Explanation

Let equation of line `L_(1)` be y = mx. Intercepts made by `L_(1)` and `L_(2)` on the circle will be equal i.e., `L_(1) and L_(2)` are at the same distance from the centre of the circle,
Centre of the given circle is (1/2, -3/2). Therefore,
`(|1//2-3//2-1|)/(sqrt(1+1))= |((m)/(2)+(3)/(2))/(sqrt(m^(2)-1))|rArr(2)/(sqrt2)=
(|m+3|)/(2sqrt(m^(2)+1))`
`rArr 8 m^(2)+8=m^(2)+6m+9`
`rArr 7m^(2)-6m-1=0rArr(7m+1)(m-1)=0`
`rArr m = -(1)/(7), m = 1`
thus, two chords are x + 7y =0
and x -y =0.
therefore, (b) and (c) are correct answers.