Question

If the circles $x^2+y^2-10x+16y+89-r^2=0$ and $x^2+y^2+6x-14y+42=0$ have common points, then the number of possible integral values of $r$ is equal to

• A. $13$
• B. $14$
• C. $15$
• D. $18$

Answer 1

Answer: (D)

Centre & radius of $x^2+y^2-10x+16y+89-r^2=0$ are $\left(5,-8\right)$ & $\left|r\right|$ respectively
Centre & radius of $x^2+y^2+6x-14y+42=0$ are $\left(-3,7\right)$ & $4$ respectively
Distance between the centres of circles is $\sqrt{8^2+15^2}=17$
$\Rightarrow 17-4\le \left|r\right|\le 17+4$
$\Rightarrow 13\le \left|r\right|\le 21$
$\Rightarrow r\in \left\{\pm 21,\pm 20,\pm 19,\pm 18,\pm 17,\pm 16,\pm 15,\pm 14,\pm 13\right\}$
Hence, the number of possible values of $r$ is equal to $18$