Understanding Concentric Circles
Two circles are said to be concentric if they share the exact same center point but have different radii. In the Cartesian coordinate system, the general equation of a circle is given by:
$x^2 + y^2 + 2gx + 2fy + c = 0$
For two circles to be concentric, the coefficients of $x$ and $y$ must be identical, as they determine the center $(h, k) = (-g, -f)$. Therefore, a circle concentric to $x^2 + y^2 + 2gx + 2fy + c = 0$ will always take the form:
$x^2 + y^2 + 2gx + 2fy + k = 0$
where $k$ is an unknown constant we need to determine.
Solving the Problem
Given the equation of the circle: $x^2 + y^2 - 8x + 12y + 15 = 0$
Step 1: Define the concentric form
Since the new circle must be concentric, it will have the same $x$ and $y$ terms. We can represent the required circle as: $x^2 + y^2 - 8x + 12y + K = 0$ where $K$ is a constant to be determined.
Step 2: Use the passing point
We are told the circle passes through the point $(5, 4)$. This means that when we substitute $x = 5$ and $y = 4$ into our new equation, the equation must hold true. Let's substitute:
$(5)^2 + (4)^2 - 8(5) + 12(4) + K = 0$
Step 3: Solve for K
Now, calculate the values: $25 + 16 - 40 + 48 + K = 0$ $41 - 40 + 48 + K = 0$ $1 + 48 + K = 0$ $49 + K = 0$ $K = -49$
Step 4: Write the final equation
Substitute the value of $K$ back into the concentric form equation from Step 1: $x^2 + y^2 - 8x + 12y - 49 = 0$
Conclusion
The equation of the circle concentric to $x^2 + y^2 - 8x + 12y + 15 = 0$ and passing through the point $(5, 4)$ is:
$x^2 + y^2 - 8x + 12y - 49 = 0$