Show that the equations of eth straight lines passing through the point `(3,-2)` and inclined at `60^0` to the line `sqrt(3)x+y=1a r ey+2=0a n dy-sqrt(3)x+2+3sqrt(3)=0.`

Options

A. `y+2=0,sqrt3x-y-2-3sqrt3=0`
B. `x-2=0,sqrt3x-y+2+3sqrt3=0`
C. `sqrt3x-y-2-3sqrt3=0`
D. none of above

Answer

A

Explanation

So, the given point A is (3,-2)
So, the equation of line `sqrt3x+y=1`
`rArry=-sqrt3x+1`
`therefore` Slope, `m_1=-sqrt3`
Let slope of the required line be `m_2`.
`therefore tan theta =|(-sqrt3-m_2)/(1-sqrt3m_2)|` , `[because tan theta =|(m_1-m_2)/(1+m_1m_2)|]`
`rArr tan 60^@ =pm((-sqrt3-m_2)/(1-sqrt3m_2))`
`rArrsqrt3=((-sqrt3-m_2)/(1-sqrt3m_2))` [Taking positive sign]
`rArr sqrt3-3m_2=-sqrt3-m_2`
`rArr2sqrt3=2m_2`
`rArrm_2=sqrt3`
`therefore` Equation of line passing through `(3,-2)` is `y+2=sqrt3(x-3)`
`y-sqrt3x+2+3sqrt3=0`
`rArrsqrt3-y-2-3sqrt=0`
[Taking negative sign from Eq.(i)]
`rArrsqrt3-3m_2=sqrt3+m_2`
`rArrm_2=0`
`therefore` The equation of line is `y+2=0(x-3)`
`rArry+2=0`
So, the required equation of lines are `sqrt3x-y-2-sqrt(3)=0` and `y+2=0`.