Question

Choose the incorrect statement about the two circles whose equations are given below:
$x^2+y^2-10x-10y+41=0$ and $x^2+y^2-16x-10y+80=0$

• A. Distance between two centres is the average of radii of both the circles.
• B. Circles have two intersection points.
• C. Both circles centres lie inside region of one another.
• D. Both circles pass through the centre of each other.

Answer 1

The correct option is C Both circles centres lie inside region of one another.

Given circles are
$S_1:x^2+y^2-10x-10y+41=0S_2:x^2+y^2-16x-10y+80=0$
Center and radius of the circles
$C_1=(5,5),r_1=3C_2=(8,5),r_2=3$

Distance between the centers is
$3=\frac{r_1+r_2}{2}$

$C_1{C}_2=3,r_1+r_2=6,|r_1-r_2|=0\Rightarrow r_1+r_2>C_1{C}_2>|r_1-r_2|$
The circles intersect at two points.

Position of $C_1(5,5)$ in $S_2=0$
$25+25-80-50+80=0$
Position of $C_2(8,5)$ in $S_1=0$
$64+25-80-50+41=0$
Both circles pass through the centre of each other.