`x^2+y^2-10x-10y+41=0 and x^2+y^2-16x-10y+80=0` are two circle which of the following is NOT correct

Options

A. Distance between centers is equal to average of radii
B. Both circles passes through centres of each other
C. Centers of each circle is contained in other circle
D. Both circle intersect at 2 point

Answer

To solve the problem, we need to analyze the two given equations of circles and determine which statement about them is NOT correct.

### Step 1: Identify the equations of the circles

The two equations of the circles are:
1. \( x^2 + y^2 - 10x - 10y + 41 = 0 \) (Circle 1)
2. \( x^2 + y^2 - 16x - 10y + 80 = 0 \) (Circle 2)

### Step 2: Rewrite the equations in standard form

To find the center and radius of each circle, we will rewrite each equation in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \).

**For Circle 1:**
\[ x^2 - 10x + y^2 - 10y + 41 = 0 \]
Completing the square:
\[ (x^2 - 10x + 25) + (y^2 - 10y + 25) = 9 \]
This simplifies to:
\[ (x - 5)^2 + (y - 5)^2 = 3^2 \]
Thus, the center \( C_1 \) is \( (5, 5) \) and the radius \( R_1 = 3 \).

**For Circle 2:**
\[ x^2 - 16x + y^2 - 10y + 80 = 0 \]
Completing the square:
\[ (x^2 - 16x + 64) + (y^2 - 10y + 25) = 9 \]
This simplifies to:
\[ (x - 8)^2 + (y - 5)^2 = 3^2 \]
Thus, the center \( C_2 \) is \( (8, 5) \) and the radius \( R_2 = 3 \).

### Step 3: Calculate the distance between the centers

The distance \( d \) between the centers \( C_1(5, 5) \) and \( C_2(8, 5) \) is:
\[ d = \sqrt{(8 - 5)^2 + (5 - 5)^2} = \sqrt{3^2} = 3 \]

### Step 4: Analyze the relationship between the circles

To determine the relationship between the two circles, we check the following conditions:
1. If \( d < R_1 + R_2 \) (the circles intersect).
2. If \( d = R_1 + R_2 \) (the circles touch externally).
3. If \( d > R_1 + R_2 \) (the circles are separate).
4. If \( d < |R_1 - R_2| \) (one circle is inside the other without touching).

Here, we have:
- \( R_1 + R_2 = 3 + 3 = 6 \)
- \( d = 3 \)

Since \( d < R_1 + R_2 \), the circles intersect at two points.

### Step 5: Check if the centers lie on each other's circles

To check if the center of Circle 1 lies on Circle 2:
Substituting \( C_1(5, 5) \) into the equation of Circle 2:
\[ (5 - 8)^2 + (5 - 5)^2 = 9 \implies 9 + 0 = 9 \quad \text{(True)} \]

To check if the center of Circle 2 lies on Circle 1:
Substituting \( C_2(8, 5) \) into the equation of Circle 1:
\[ (8 - 5)^2 + (5 - 5)^2 = 9 \implies 9 + 0 = 9 \quad \text{(True)} \]

### Conclusion

Both centers lie on each other's circles, and both circles intersect at two points. Therefore, we can conclude that the statement that is NOT correct is likely one that suggests they do not intersect or that one is not contained within the other.