Question

If two circles ${(x-1)}^2+{(y-3)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect in two distinct points then ___________.

• A. $2<r<8$
• B. $r<2$
• C. $r=2$
• D. $r>2$

Answer 1

The correct option is A $2<r<8$

Let $d$ be the distance between the centres of two circles of radii $r_1$ and $r_2$.

These circle intersect at two distinct point if

$|r_1–r_2|<d<r_1+r_2$

Here, the radii of the two circles are $r$ and $3$ and distance between the centre is $5$.

Thus,$|r–3|<5<r+3$

$\Rightarrow –2<r<8$ and $r>2$

$\Rightarrow 2<r<8$