If a circle is concentric with the circle `x^(2)+y^(2)-4x-6y+9=0` and passes through the point (-4,-5) then its equation is

Options

A. `x^(2)+y^(2)+4x+6y-87=0`
B. `x^(2)+y^(2)+4x+6y+87=0`
C. `x^(2)+y^(2)-4x-6y-87=0`
D. `x^(2)+y^(2)-4x-6y+87=0`

Answer

To solve the problem, we need to determine the equation of a circle that is concentric with the given circle and passes through the point (-4, -5).

### Step 1: Identify the center of the given circle

The equation of the given circle is:

\[ x^2 + y^2 - 4x - 6y + 9 = 0 \]

We can rewrite this in standard form by completing the square.

1. Rearranging the equation:
\[ x^2 - 4x + y^2 - 6y + 9 = 0 \]

2. Completing the square for \(x\) and \(y\):
- For \(x^2 - 4x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
- For \(y^2 - 6y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

3. Substituting back into the equation:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 + 9 = 0 \]
Simplifying gives:
\[ (x - 2)^2 + (y - 3)^2 - 4 = 0 \]
Thus,
\[ (x - 2)^2 + (y - 3)^2 = 4 \]

The center of the circle is \((2, 3)\).

### Step 2: Determine the radius of the new circle

Since the new circle is concentric with the given circle, it will have the same center \((2, 3)\). The radius of the new circle can be found using the distance from the center to the point (-4, -5).

1. Calculate the distance (radius \(r\)):
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (-4, -5)\).

Substituting the values:
\[ r = \sqrt{((-4) - 2)^2 + ((-5) - 3)^2} \]
\[ = \sqrt{(-6)^2 + (-8)^2} \]
\[ = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 3: Write the equation of the new circle

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

Substituting \(h = 2\), \(k = 3\), and \(r = 10\):

\[ (x - 2)^2 + (y - 3)^2 = 10^2 \]
\[ (x - 2)^2 + (y - 3)^2 = 100 \]

### Final Answer

The equation of the circle is:

\[ (x - 2)^2 + (y - 3)^2 = 100 \]

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