Question

If the circles \((x+1)^{2}+(y+2)^{2}=r^{2}\) and \(x^{2}+y^{2}-4 x-4 y+4=0\) intersect at exactly two distinct points, then

(1) \(5<\mathrm{r}<9\)

(2) \(0<\mathrm{r}<7\)

(3) \(3<\mathrm{r}<7\)

(4) \(\frac{1}{2}<\mathrm{r}<7\)

Answer 1

Correct option is (3) \(3<\mathrm{r}<7\)

If two circles intersect at two distinct points

\(\Rightarrow\left|\mathrm{r}_{1}-\mathrm{r}_{2}\right|<\mathrm{C}_{1} \mathrm{C}_{2}<\mathrm{r}_{1}+\mathrm{r}_{2}\)

\(|\mathrm{r}-2|<\sqrt{9+16}<\mathrm{r}+2\)

\(|\mathrm{r}-2|<5\text{ and }\mathrm{r}+2>5\)

\(-5<\mathrm{r}-2<5\text{ and }\mathrm{r}>3 \quad ....(2)\)

\(-3<\mathrm{r}<7 \quad ....(1)\)

From (1) and (2)

\(3<\mathrm{r}<7\)