The equation of the circle with centre (2,3) and distance between (0,0) and (3,4) as radius is

Options

A. `x^(2)+y^(2)-4x-6y-12=0`
B. `x^(2)+y^(2)-4x-6y+12=0`
C. `x^(2)+y^(2)-4x-6y-1=0`
D. `x^(2)+y^(2)-2x-3y-12=0`

Answer

To find the equation of the circle with center (2, 3) and radius equal to the distance between the points (0, 0) and (3, 4), we will follow these steps:

### Step 1: Find the radius using the distance formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, we need to find the distance between the points (0, 0) and (3, 4):
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

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

In this case, the center is \( (2, 3) \) and the radius is \( 5 \). Therefore, we can substitute these values into the equation:
\[ (x - 2)^2 + (y - 3)^2 = 5^2 \]
\[ (x - 2)^2 + (y - 3)^2 = 25 \]

### Step 3: Expand the equation
Now, we will expand the left side of the equation:
\[ (x - 2)^2 = x^2 - 4x + 4 \]
\[ (y - 3)^2 = y^2 - 6y + 9 \]

Combining these, we have:
\[ x^2 - 4x + 4 + y^2 - 6y + 9 = 25 \]

### Step 4: Simplify the equation
Now, combine like terms:
\[ x^2 + y^2 - 4x - 6y + (4 + 9) = 25 \]
\[ x^2 + y^2 - 4x - 6y + 13 = 25 \]

Subtract 25 from both sides:
\[ x^2 + y^2 - 4x - 6y + 13 - 25 = 0 \]
\[ x^2 + y^2 - 4x - 6y - 12 = 0 \]

### Final Equation
Thus, the equation of the circle is:
\[ x^2 + y^2 - 4x - 6y - 12 = 0 \]

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