Understanding Concentric Circles
Two circles are said to be concentric if they share the same center point. Because the center is the same, their equations will look identical in terms of the $x$ and $y$ variables—the only difference will be the constant term.
In the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$, the center is given by the coordinates $(-g, -f)$. If two circles are concentric, their $g$ and $f$ values must be identical.
The Problem
We need to find the equation of a circle that is concentric with: $$x^2 + y^2 - 8x + 12y + 15 = 0$$ and passes through the point $(5, 4)$.
Step-by-Step Solution
Step 1: Define the concentric circle
Since the new circle shares the same center as the given circle, we keep the $x$ and $y$ terms the same and replace the constant term with a variable, let's call it $k$: $$x^2 + y^2 - 8x + 12y + k = 0$$
Step 2: Use the given point to solve for $k$
We know the circle passes through $(5, 4)$. This means that when $x = 5$ and $y = 4$, the equation must hold true. Substitute these values into our equation: $$(5)^2 + (4)^2 - 8(5) + 12(4) + k = 0$$
Perform the arithmetic: $$25 + 16 - 40 + 48 + k = 0$$ $$41 - 40 + 48 + k = 0$$ $$1 + 48 + k = 0$$ $$49 + k = 0$$ $$k = -49$$
Step 3: State the final equation
Now substitute the value of $k$ back into our placeholder equation from Step 1: $$x^2 + y^2 - 8x + 12y - 49 = 0$$
Summary
The equation of the circle concentric to $x^2 + y^2 - 8x + 12y + 15 = 0$ passing through $(5, 4)$ is: $x^2 + y^2 - 8x + 12y - 49 = 0$