Understanding the Problem
To find the equation of a circle, we typically need two pieces of information: the coordinates of its center $(h, k)$ and its radius $r$. The standard form of a circle's equation is:
$(x - h)^2 + (y - k)^2 = r^2$
In this problem, we are given:
- The center $(h, k) = (-3, -4)$
- The condition that the circle touches the x-axis.
Visualizing the Geometry
When a circle touches the x-axis, the perpendicular distance from the center $(h, k)$ to the x-axis must be equal to the radius of the circle.
- The x-axis is the line $y = 0$.
- The distance from any point $(h, k)$ to the x-axis is simply the absolute value of its y-coordinate, $|k|$.
- Therefore, $r = |k|$.
Given the center $(-3, -4)$, the radius is: $r = |-4| = 4$
Step-by-Step Solution
1. Identify the center and radius
- Center $(h, k) = (-3, -4)$
- Radius $r = 4$
2. Substitute into the standard equation
Using the formula $(x - h)^2 + (y - k)^2 = r^2$, we plug in our values: $(x - (-3))^2 + (y - (-4))^2 = 4^2$
3. Simplify the equation
$(x + 3)^2 + (y + 4)^2 = 16$
4. Optional: Expand to General Form
If you need the general form, expand the squares: $(x^2 + 6x + 9) + (y^2 + 8y + 16) = 16$ $x^2 + y^2 + 6x + 8y + 25 = 16$ $x^2 + y^2 + 6x + 8y + 9 = 0$
Conclusion
The equation of the circle is $(x + 3)^2 + (y + 4)^2 = 16$ (or $x^2 + y^2 + 6x + 8y + 9 = 0$).
Pro-Tip: If a circle touches the y-axis instead, the radius would be the absolute value of the x-coordinate, $r = |h|$. Always check which axis the circle is tangent to!