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 centers is the average radii of both the circles.
• B. Circles have two intersection points.
• C. Both circles’ centers lie inside the region of one another.
• D. Both circles pass through the center of each other.

Answer 1

The correct option is C

Both circles’ centers lie inside the region of one another.

Explanation for the correct option:

Step1. Given equations of the circles :

given equation of the circles

$$\begin{gathered} x^2+y^2–10x–10y+41=0.........\left(1\right) \\ and{x}^2+y^2–16x–10y+80=0..........\left(2\right) \end{gathered}$$

Compared with the general equation of the circle

$x^2+y^2+2gx+2fy+c=0$

whose center is $(-g,-f)$and radius is $\sqrt{g^2+f^2-c}$ .

then the center of the circle (1) $C_1=(5,5)$and the center of the circle (2) $C_2=(8,5)$

the radius of the circle (1) $r_1=\sqrt{5^2+5^2-41}=\sqrt{9}=3$ and radius of the circle (2) $r_2=\sqrt{8^2+5^2-80}=\sqrt{9}=3$

Step 2. Check the position of the centers of both circles:

The position of$C_1=(5,5)$ in circle (2)

$=25+25–80–50+80=0$

The position of$C_2=(8,5)$ in circle (1)

$=64+25–80–50+41=0$

Thus, both circles’ centers lie inside the region of one another.

Hence, the correct option is (C).