Question

$A$ is a point on the circle $x^2+y^2-8x-8y+28=0$, $B$ is a point on the circle $x^2+y^2-2x-3=0$. If the distance between $A$ and $B$ is $d$, then

• A. $1\le d\le 9$
• B. $2\le d\le 8$
• C. $1\le d\le 5$
• D. $3\le d\le 6$

Answer 1

The correct option is A $1\le d\le 9$

$S_1:x^2+y^2-8x-8y+28=0$
$\Rightarrow C_1\equiv (4,4)$ and $r_1=2$

$S_2:x^2+y^2-2x-3=0$
$\Rightarrow C_2\equiv (1,0)$ and $r_2=2$

Distance between Centres, $C_{1}and{C}_2$, is

$dis(C_1,C_2)=5$

which is greater than $r_1+r_2=4$,

$\therefore S_1$ and $S_2$ do not intersect each other.

Distance between point $A$ and $B$ is $d$.
$\therefore d_{\text{max}}=dis(C_1,C_2)+(r_1+r_2)$
$=5+4=9$
$d_{\text{min}}=dis(C_1,C_2)-(r_1+r_2)$
$=5-2-2=1$

$\Rightarrow 1\le d\le 9$

Hence, option A.