A circle touches the line `y=x` at point `(4,4)` on it. The length of the chord on the line `x+y=0` is `6sqrt2`. Then one of the possible equation of the circle is

Options

A. `x^(2)+y^(2)+x-y+30=0`
B. `x^(2)+y^(2)+2x-18y+32=0`
C. `x^(2)+y^(2)+2x+18y+32=0`
D. `x^(2)+y^(2)-2x-22y+32=0`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have a circle that touches the line \( y = x \) at the point \( (4, 4) \). The length of the chord on the line \( x + y = 0 \) is \( 6\sqrt{2} \). We need to find one possible equation of the circle.

### Step 2: Determine the Center of the Circle
Since the circle touches the line \( y = x \) at the point \( (4, 4) \), the center of the circle must lie on the line perpendicular to \( y = x \) that passes through \( (4, 4) \). The slope of \( y = x \) is \( 1 \), so the slope of the perpendicular line is \( -1 \).

Using the point-slope form of the line, we can write the equation of the line through \( (4, 4) \):
\[ y - 4 = -1(x - 4) \implies y = -x + 8 \]

### Step 3: Find the Relationship Between Center Coordinates
Let the center of the circle be \( (h, k) \). Since the center lies on the line \( y = -x + 8 \), we have:
\[ k = -h + 8 \quad \text{(1)} \]

### Step 4: Calculate the Radius
The radius \( r \) of the circle can be expressed as the distance from the center \( (h, k) \) to the point \( (4, 4) \):
\[ r = \sqrt{(h - 4)^2 + (k - 4)^2} \]

### Step 5: Find the Perpendicular Distance to the Line \( x + y = 0 \)
The distance \( d \) from the center \( (h, k) \) to the line \( x + y = 0 \) is given by:
\[ d = \frac{|h + k|}{\sqrt{2}} \]

### Step 6: Relate the Chord Length to the Radius
The length of the chord on the line \( x + y = 0 \) is given as \( 6\sqrt{2} \). The relationship between the radius \( r \) and the chord length \( L \) is:
\[ L = 2 \sqrt{r^2 - d^2} \]
Substituting \( L = 6\sqrt{2} \):
\[ 6\sqrt{2} = 2 \sqrt{r^2 - d^2} \]
Dividing both sides by 2:
\[ 3\sqrt{2} = \sqrt{r^2 - d^2} \]
Squaring both sides:
\[ 18 = r^2 - d^2 \quad \text{(2)} \]

### Step 7: Substitute for \( r \) and \( d \)
From equation (1), substitute \( k \) in terms of \( h \) into the expression for \( r \):
\[ r = \sqrt{(h - 4)^2 + (-h + 8 - 4)^2} = \sqrt{(h - 4)^2 + (-h + 4)^2} \]
Simplifying:
\[ r = \sqrt{(h - 4)^2 + (4 - h)^2} = \sqrt{(h - 4)^2 + (h - 4)^2} = \sqrt{2(h - 4)^2} = \sqrt{2} |h - 4| \]

Now substitute \( r \) and \( d \) into equation (2):
\[ 18 = 2(h - 4)^2 - \frac{(h + (-h + 8))^2}{2} = 2(h - 4)^2 - \frac{(8)^2}{2} = 2(h - 4)^2 - 32 \]
Thus:
\[ 18 + 32 = 2(h - 4)^2 \implies 50 = 2(h - 4)^2 \implies (h - 4)^2 = 25 \implies |h - 4| = 5 \]
This gives us two cases:
1. \( h - 4 = 5 \implies h = 9 \)
2. \( h - 4 = -5 \implies h = -1 \)

### Step 8: Find Corresponding \( k \)
Using \( k = -h + 8 \):
1. If \( h = 9 \), then \( k = -9 + 8 = -1 \).
2. If \( h = -1 \), then \( k = 1 + 8 = 9 \).

### Step 9: Write the Equation of the Circle
For \( (h, k) = (9, -1) \):
\[ (x - 9)^2 + (y + 1)^2 = r^2 \]
Calculating \( r \):
\[ r = \sqrt{2} \cdot 5 = 5\sqrt{2} \]
Thus, the equation becomes:
\[ (x - 9)^2 + (y + 1)^2 = 50 \]

For \( (h, k) = (-1, 9) \):
\[ (x + 1)^2 + (y - 9)^2 = 50 \]

### Final Answer
One possible equation of the circle is:
\[ (x - 9)^2 + (y + 1)^2 = 50 \]
or
\[ (x + 1)^2 + (y - 9)^2 = 50 \]