The Circle

The Circle

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is the radius.

Standard Equation

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

Standard Equation

$(x - h)^2 + (y - k)^2 = r^2$

If the center is at the origin $(0, 0)$, the equation simplifies to: $x^2 + y^2 = r^2$

General Equation

The general form of the equation of a circle is obtained by expanding the standard equation:

General Equation

$x^2 + y^2 + Dx + Ey + F = 0$

Where $D, E, F$ are constants.

Finding the Center and Radius

To find the center $(h, k)$ and radius $r$ from the general equation, we can complete the square for both $x$ and $y$ terms.

Alternatively, using formulas derived from expanding $(x-h)^2 + (y-k)^2 = r^2$:

• $h = -D/2$
• $k = -E/2$
• $r = \sqrt{h^2 + k^2 - F}$

Tangent to a Circle

A line is tangent to a circle if it touches the circle at exactly one point. The radius at the point of tangency is perpendicular to the tangent line.

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Solved Problems