If the lines `2x-3y=5and3x-4y=7` are two diameters of a circle of radius 7 , then the equation of the circle is

Options

A. `x^(2)+y^(2)+2x-4y-47=0`
B. `x^(2)+y^(2)=49`
C. `x^(2)+y^(2)-2x+2y-47=0`
D. `x^(2)+y^(2)=17`

Answer

To find the equation of the circle given that the lines \(2x - 3y = 5\) and \(3x - 4y = 7\) are diameters of the circle with a radius of 7, we can follow these steps:

### Step 1: Find the intersection point of the two lines
We need to solve the equations of the lines to find their intersection point, which will be the center of the circle.

The equations are:
1. \(2x - 3y = 5\) (Equation 1)
2. \(3x - 4y = 7\) (Equation 2)

To solve these equations, we can eliminate one variable. Let's eliminate \(x\).

Multiply Equation 1 by 3 and Equation 2 by 2 to make the coefficients of \(x\) the same:

\[ 3(2x - 3y) = 3(5) \implies 6x - 9y = 15 \quad \text{(Equation 3)} \]
\[ 2(3x - 4y) = 2(7) \implies 6x - 8y = 14 \quad \text{(Equation 4)} \]

### Step 2: Subtract the equations
Now, we subtract Equation 4 from Equation 3:

\[ (6x - 9y) - (6x - 8y) = 15 - 14 \]
This simplifies to:
\[ -y = 1 \implies y = -1 \]

### Step 3: Substitute \(y\) back to find \(x\)
Now that we have \(y = -1\), we can substitute this value back into either Equation 1 or Equation 2 to find \(x\). Let's use Equation 1:

\[ 2x - 3(-1) = 5 \]
\[ 2x + 3 = 5 \implies 2x = 2 \implies x = 1 \]

### Step 4: Determine the center of the circle
The intersection point, which is the center of the circle, is \((1, -1)\).

### Step 5: Write the equation of the circle
The general equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, \(h = 1\), \(k = -1\), and \(r = 7\). Therefore, the equation becomes:
\[ (x - 1)^2 + (y + 1)^2 = 7^2 \]
\[ (x - 1)^2 + (y + 1)^2 = 49 \]

### Step 6: Expand the equation
Now we can expand the equation:
\[ (x - 1)^2 + (y + 1)^2 = 49 \]
Expanding gives:
\[ (x^2 - 2x + 1) + (y^2 + 2y + 1) = 49 \]
Combining like terms:
\[ x^2 + y^2 - 2x + 2y + 2 = 49 \]
Subtracting 49 from both sides:
\[ x^2 + y^2 - 2x + 2y - 47 = 0 \]

### Final Equation of the Circle
Thus, the equation of the circle is:
\[ x^2 + y^2 - 2x + 2y - 47 = 0 \]

---