consider a family of circles passing through two fixed points `S(3,7)` and `B(6,5)`. If the common chords of the circle `x^(2)+y^(2)-4x-6y-3=0` and the members of the family of circles pass through a fixed point (a,b), then

Options

A. `a+3b=25`
B. `3b-a=21`
C. `3a+b=25`
D. `3a-b=21`

Answer

A, B

Explanation

Family of circle `S+lamdaL=0`
`S_(1)-=(x-3)(x-6)+(y-7)(y-5)+lamda(2x+3-27)=0`
`x^(2)+y^(2)+x(2lamda-9)+y(3lamda-12)+(53-27lamda)=0`
common chord of family of circle and given circle is `S_(1)-S_(2)`
`=0(S_(22)` given circle)
`(-5x-6y+56)+lamda(2x+3y-27)=0`
Which represents family of lines passing through the point
`(2,(23)/(3))impliesa=2,b=(23)/(3)`