If the two circles $(x-1)^2+(y-3)^2=r^2$ and $x^2+y^2-8 x+2 y+8=0$ intersect in two different points, then what can we conclude about $r$ ?

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Aicharya · Accepted Answer

As circles intersects in two distinct points

So, $\left|r_1-r_2\right| < \left|C_1 C_2\right| < r_1+r_2$ ...(i) Here, $\quad C_1=(1,3), C_2=(4,-1)$ $\begin{aligned} \therefore \quad C_1 C_2 & =\sqrt{(4-1)^2+(-1-3)^2} \\ & =\sqrt{9+16}=5 \end{aligned}$ $\text {Also, } \begin{aligned} r_2 & =\sqrt{g^2+f^2-C} \\ & =\sqrt{42+1^2-8} \\ & =3 \end{aligned}$ Hence, from Eq. (i), we have $2 < r < 8$.

If the two circles \((x-1)^2+(y-3)^2=r^2\) and \(x^2+y^2-8 x+2 y+8=0\) intersect in two different points,…

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