Equation of the circle with radius 10 and whose two diameters are `x+y=6 and x+2y=4" is "`

Options

A. `x^(2)+y^(2)+16x-4y-32=0`
B. `x^(2)+y^(2)-16x+4y-32=0`
C. `x^(2)+y^(2)+16x-4y+32=0`
D. None

Answer

To find the equation of the circle with a radius of 10 and diameters given by the lines \(x + y = 6\) and \(x + 2y = 4\), we can follow these steps:

### Step 1: Find the intersection of the two diameters
The center of the circle is located at the intersection point of the two diameters. We can solve the equations simultaneously.

1. **Equation 1**: \(x + y = 6\)
2. **Equation 2**: \(x + 2y = 4\)

Subtract Equation 1 from Equation 2:
\[ (x + 2y) - (x + y) = 4 - 6 \]
This simplifies to:
\[ y = -2 \]

### Step 2: Substitute \(y\) back to find \(x\)
Now, substitute \(y = -2\) into Equation 1 to find \(x\):
\[ x + (-2) = 6 \]
This simplifies to:
\[ x = 8 \]

### Step 3: Determine the center and radius
Now we have the center of the circle:
\[ C(8, -2) \]
The radius \(r\) is given as 10.

### Step 4: Write the 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 \(h = 8\), \(k = -2\), and \(r = 10\):
\[ (x - 8)^2 + (y + 2)^2 = 10^2 \]
This simplifies to:
\[ (x - 8)^2 + (y + 2)^2 = 100 \]

### Step 5: Expand the equation
Now, we can expand the equation:
\[ (x - 8)^2 = x^2 - 16x + 64 \]
\[ (y + 2)^2 = y^2 + 4y + 4 \]
Combining these, we get:
\[ x^2 - 16x + 64 + y^2 + 4y + 4 = 100 \]
This simplifies to:
\[ x^2 + y^2 - 16x + 4y + 68 - 100 = 0 \]
Thus:
\[ x^2 + y^2 - 16x + 4y - 32 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 16x + 4y - 32 = 0 \]

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