Question

Consider a family of circles passing through two given points $A(3,7)$ and $B(6,5)$.
Number of circles, which belong to the family and also touch X-axis, is:

• A. 0
• B. 1
• C. 2
• D. Infinite

Answer 1

Answer: (C)

Equation of the line joining the points $A(3,7)$ and $B$ $(6,5)$ is
$y-5=\frac{7-5}{3-6}(x-6)\text{ or }2x+3y-27=0$
Equation of the family of circles through the points $A$ and $B$ is given by-$S=(x-3)(x-6)+(y-7)(y-5)+\lambda (2x+3y-27)=0$
or $S=x^2+y^2+(2\lambda -9)x+(3\lambda -12)y+53-27\lambda =0$..(i)
The equation (1) represents a circle touching the axis of ' $x$ ' if
${\left(\frac{2\lambda -9}{2}\right)}^2-(33-27\lambda )=0$
or $4{\lambda }^2+72\lambda -51=0$
Clearly, the above equation gives two distinct values of $\lambda$, so two circles are possible.