The circle with centre (1, 2) cuts the circle `x^2 + y^2 + 14x - 16y + 77 = 0` orthogonally then its radius is

Options

A. 4
B. 8
C. 3
D. 2

Answer

To solve the problem, we need to find the radius of the circle with center (1, 2) that cuts the given circle orthogonally. The given circle is represented by the equation:

\[ x^2 + y^2 + 14x - 16y + 77 = 0 \]

### Step 1: Identify the center and radius of the given circle

The general form of a circle is:

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

From the given equation, we can identify:

- \( 2g = 14 \) which gives \( g = 7 \)
- \( 2f = -16 \) which gives \( f = -8 \)
- \( c = 77 \)

Thus, the center \( C_1 \) of the given circle is:

\[ C_1 = (-g, -f) = (-7, 8) \]

To find the radius \( r_1 \), we use the formula:

\[ r_1 = \sqrt{(-g)^2 + (-f)^2 - c} \]

Calculating \( r_1 \):

\[ r_1 = \sqrt{(-7)^2 + (8)^2 - 77} = \sqrt{49 + 64 - 77} = \sqrt{36} = 6 \]

### Step 2: Calculate the distance between the centers of the circles

Let \( C_2 \) be the center of the circle we are looking for, which is \( (1, 2) \). We need to find the distance \( d \) between \( C_1 \) and \( C_2 \):

\[ d = \sqrt{(-7 - 1)^2 + (8 - 2)^2} = \sqrt{(-8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Step 3: Use the orthogonality condition to find the radius of the second circle

For two circles to cut orthogonally, the following condition must hold:

\[ r_1^2 + r_2^2 = d^2 \]

Substituting the known values:

\[ 6^2 + r_2^2 = 10^2 \]

This simplifies to:

\[ 36 + r_2^2 = 100 \]

Solving for \( r_2^2 \):

\[ r_2^2 = 100 - 36 = 64 \]

Taking the square root gives us the radius \( r_2 \):

\[ r_2 = \sqrt{64} = 8 \]

### Conclusion

The radius of the circle with center (1, 2) that cuts the given circle orthogonally is:

\[ \boxed{8} \]