The middle point of chord `x+3y=2` of the conic `x^(2)+xy-y^(2)=1,` is

Options

A. `(5, -1)`
B. `(1,1)`
C. `(2, 0)`
D. `(-1,1)`

Answer

D

Explanation

The point of intersection of `x+3y=2and x^(2)+xy-y^(2)=1` is given by
`(2-3y)^(2)+(2-3y)y-y^(2)=1`
`implies4+9y^(2)-12y+2y-3y^(2)-y^(2)=1`
`implies5y^(2)-10y+3=0`
`thereforey=(10pmsqrt(100-60))/(2xx5)=(10pmsqrt40)/(10)`
`therefoer x=2-3 ((10pmsqrt40)/(10))`
`=(-10pmsqrt40)/(10)`
`therefore` Points of intersection are
`A(-1-(sqrt40)/(10),1+(sqrt40)/(10))`
`and B(-1+(sqrt40)/(10),-(sqrt40)/(10)).`
`therefore` Mid-point of AB is (-1,1).`