Question

Consider a family of circles passing through two fixed points $A(3,7)$ and $B(6,5)$. The chord, in which the circle $x^2+y^2-4x-6y-3=0$ cuts each member of the family of circles, passes through a fixed point $(a,b)$. Find the numerical value of $a+3b$.

• A. 3
• B. 13
• C. 25
• D. 35

Answer 1

Answer: (C)

25

Family of circles is $(x-3)(x-6)+(y-7)(y-5)+\lambda (2x+3y-27)=0$
i.e. $x^2+y^2+x(2\lambda -9)+y(3\lambda -12)+(53-27\lambda )=0$
Common chord of family of circles and the given circle is
$(-5x-6y+56)+\lambda (2x+3y-27)=0$
which represents family of line passing through point of intersection of the lines
$-5x-6y+56=0$ and $2x+3y-27=0$
The point of intersection is $[2,\frac{23}{3}]$
$a=2$
$b=\frac{23}{3}$
$a+3b=2+23=25$