The equation of a circle which touches the line `x +y= 5` at `N(-2,7)` and cuts the circle `x^2 + y^2 + 4x-6y + 9 = 0` orthogonally, is

Options

A. `x^(2) + y^(2) + 7x - 12y +m 38 =0`
B. `x^(2) + y^(2) + 7x + 11y + 38 =0`
C. `x^(2) + y^(2) + 7x - 11y - 38 =0`
D. `x^(2) + y^(2) -7x -11y + 39 =0`

Answer

To find the equation of the circle that touches the line \( x + y = 5 \) at the point \( N(-2, 7) \) and cuts the circle given by \( x^2 + y^2 + 4x - 6y + 9 = 0 \) orthogonally, we can follow these steps:

### Step 1: Identify the center and radius of the circle
Since the circle touches the line \( x + y = 5 \) at the point \( N(-2, 7) \), we can consider the center of the circle to be at \( N(-2, 7) \).

### Step 2: Write the general equation of the circle
The general equation of a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = -2 \) and \( k = 7 \):
\[ (x + 2)^2 + (y - 7)^2 = r^2 \]

### Step 3: Find the distance from the center to the line
The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( x + y - 5 = 0 \) (where \( A = 1, B = 1, C = -5 \)):
\[ d = \frac{|1(-2) + 1(7) - 5|}{\sqrt{1^2 + 1^2}} = \frac{|-2 + 7 - 5|}{\sqrt{2}} = \frac{|0|}{\sqrt{2}} = 0 \]
Since the circle touches the line, the radius \( r \) must equal the distance \( d \).

### Step 4: Find the orthogonality condition
The circles are orthogonal if:
\[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
where \( g_1, f_1, c_1 \) are coefficients from the first circle and \( g_2, f_2, c_2 \) from the second circle.

### Step 5: Identify coefficients of the given circle
The given circle \( x^2 + y^2 + 4x - 6y + 9 = 0 \) can be rewritten in the standard form:
- Coefficients: \( g_2 = 2, f_2 = -3, c_2 = 9 \)

### Step 6: Coefficients of the unknown circle
From the equation \( (x + 2)^2 + (y - 7)^2 = r^2 \):
Expanding gives:
\[ x^2 + 4x + 4 + y^2 - 14y + 49 = r^2 \]
This can be rearranged to:
\[ x^2 + y^2 + 4x - 14y + (53 - r^2) = 0 \]
Thus, we have:
- \( g_1 = 2 \)
- \( f_1 = -7 \)
- \( c_1 = 53 - r^2 \)

### Step 7: Substitute into the orthogonality condition
Substituting into the orthogonality condition:
\[ 2(2)(2) + 2(-7)(-3) = (53 - r^2) + 9 \]
Calculating:
\[ 8 + 42 = 62 - r^2 \]
\[ 50 = 62 - r^2 \]
\[ r^2 = 12 \]

### Step 8: Final equation of the circle
Substituting \( r^2 = 12 \) back into the equation:
\[ (x + 2)^2 + (y - 7)^2 = 12 \]
Expanding gives:
\[ x^2 + 4x + 4 + y^2 - 14y + 49 = 12 \]
\[ x^2 + y^2 + 4x - 14y + 41 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 + 4x - 14y + 41 = 0 \]