A straight line through the point A `(-2,-3)` cuts the line `x+3y=9` and `x+y+1=0` at B and C respectively. If AB.AC`=20` then equation of the possible line is

Options

A. `x-y=1`
B. `x-y+1=0`
C. `3x-y+3=0`
D. `3x-y=3`

Answer

A, C

Explanation

Any point on line through A is
`(-2+rcostheta,-3+rsintheta)`
`because(-2+Abcostheta,-3+Absintheta)` liesd on `x+3y=9`
`becauseAB=(20)/((costheta+3sintheta))` similarly `AC=(4)/((costheta+sintheta))`
`because4=cos^(2)theta+4sinthetacostheta+3sin^(2)theta`
`4+4tan^(2)theta=1+4tantheta+3tan^(2)theta`
`tan^(2)theta-4tantheta+3=0`
`tantheta=1` or `tantheta=3`
`implies` required lines are
`y+3=x+2` or `y+3=3(x+2)`