Circle Calculator
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Step-by-Step Solution
About Circle Equations
In analytical geometry, a circle can be represented by different forms of equations:
Standard Form
$(x - h)^2 + (y - k)^2 = r^2$
where $(h, k)$ is the center of the circle and $r$ is the radius.
General Form
$x^2 + y^2 + Dx + Ey + F = 0$
where $D = -2h$, $E = -2k$, and $F = h^2 + k^2 - r^2$.
Converting Between Forms
From Standard to General Form:
Expand $(x - h)^2 + (y - k)^2 = r^2$ to get $x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = 0$
So $D = -2h$, $E = -2k$, and $F = h^2 + k^2 - r^2$
From General to Standard Form:
Complete the square for $x$ and $y$ terms:
$x^2 + Dx + y^2 + Ey + F = 0$
$x^2 + Dx + \frac{D^2}{4} + y^2 + Ey + \frac{E^2}{4} = -F + \frac{D^2}{4} + \frac{E^2}{4}$
$(x + \frac{D}{2})^2 + (y + \frac{E}{2})^2 = \frac{D^2 + E^2 - 4F}{4}$
So the center is $(-\frac{D}{2}, -\frac{E}{2})$ and the radius is $\sqrt{\frac{D^2 + E^2 - 4F}{4}}$
Finding a Circle Through Three Points
To find the equation of a circle passing through three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$:
- Set up a system of three equations using the general form of the circle equation.
- Solve for the coefficients $D$, $E$, and $F$.
- Convert to standard form to find the center and radius.
Applications
- Determining if a point lies on, inside, or outside a circle
- Finding the intersection points of a line and a circle
- Calculating the area and circumference of a circle
- Designing circular structures in engineering and architecture
- Modeling circular motion in physics