The two circles $x^2+y^2=r^2$ and $x^2+y^2-10 x+16=0$ intersect at two distinct points. Then which one of the following is correct?

• Correct answer
  A

  $2 \lt \mathrm{r} \lt 8$
• B

  $\mathrm{r}=2$ or $\mathrm{r}=8$
• C

  $\mathrm{r}=2$ or $\mathrm{r}=8$
• D

  $r \gt 2$

Solution

For the circle, $x^2+y^2=r^2$, centre $(0,0)$ radius $=r$ If the circle $x^2+y^2-10 x+16=0$ is compared with general form $\begin{aligned} & x^2+y^2+2 g+2 f y+c=0, \text { we get } \\ & 2 g=-10,2 f=0, c=16 \\ & \Rightarrow g=-5, f=0 \\ & \therefore \text { centre }=(+5,0) \end{aligned}$ $\text { radius }=\sqrt{\mathrm{g}^2+\mathrm{f}^2-\mathrm{c}}=\sqrt{25+0-16}=\sqrt{9}=3$ Given, two circles intersect at two distinct points,
To interset at two points, r should be greater than 2 So, $2 \lt \mathrm{r} \lt 8$

Hence, the correct answer is option: 'A'.