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
$y-5=\frac{7-5}{3-6}(x-6) \text { or } 2 x+3 y-27=0$
Equation of the family of circles through the points $A$ and $B$ is given by-$S =( x -3)( x -6)+( y -7)( y -5)+\lambda(2 x +3 y -27) = 0$
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.