Equation of a Circle

• Geometric Definition: A circle is defined as the set of all points $(x, y)$ in a plane that are at a constant distance, known as the radius ($r$), from a fixed point called the center $(h, k)$. This definition ensures that the circle is perfectly symmetrical around its central point.

• Standard Form: The most common algebraic representation is $(x - h)^2 + (y - k)^2 = r^2$. In this format, the values of $h$ and $k$ represent the horizontal and vertical shifts from the origin, while $r^2$ represents the square of the radius.

• The Origin Case: When a circle is centered at the origin $(0, 0)$, the equation simplifies significantly to $x^2 + y^2 = r^2$, which is a direct application of the Pythagorean theorem for a point on the circumference.

• Distance Formula Derivation: The equation of a circle is derived directly from the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. By setting the distance $d$ to the radius $r$ and the fixed point to $(h, k)$, squaring both sides yields the standard form.

• Pythagorean Relationship: Every point $(x, y)$ on the circle forms a right-angled triangle with the center $(h, k)$. The horizontal leg has length $|x - h|$, the vertical leg has length $|y - k|$, and the hypotenuse is the radius $r$. Thus, $(x - h)^2 + (y - k)^2 = r^2$ is simply $a^2 + b^2 = c^2$.

• Locus of Points: Algebraically, the equation represents a constraint. Only coordinate pairs that satisfy the equality lie on the boundary of the circle; points that result in a value less than $r^2$ lie inside, and those greater than $r^2$ lie outside.

Converting General Form to Standard Form

• The General Form: Circles are often presented as $x^2 + y^2 + Dx + Ey + F = 0$. To find the center and radius, you must convert this to standard form using completing the square.
• Step 1: Group Terms: Rearrange the equation to group $x$ terms together and $y$ terms together, moving the constant $F$ to the other side: $(x^2 + Dx) + (y^2 + Ey) = -F$.
• Step 2: Complete the Squares: Add $(\frac{D}{2})^2$ and $(\frac{E}{2})^2$ to both sides of the equation. This creates perfect square trinomials on the left side.
• Step 3: Factor and Simplify: Rewrite the grouped terms as $(x + \frac{D}{2})^2 + (y + \frac{E}{2})^2 = \text{constant}$. The center is $(-\frac{D}{2}, -\frac{E}{2})$ and the radius is the square root of the resulting constant.

• Circle vs. Point vs. Empty Set: In the form $(x - h)^2 + (y - k)^2 = M$, if $M > 0$, it is a circle. If $M = 0$, the 'circle' has a radius of zero and is just a single point $(h, k)$. If $M < 0$, no real points satisfy the equation, resulting in an empty set (or 'imaginary' circle).

• The Sign Trap: Always remember that the signs in the center $(h, k)$ are the opposite of the signs inside the brackets. For example, $(x + 5)^2$ means the $x$-coordinate of the center is $-5$.
• Radius vs. Radius Squared: A common mistake is to identify the constant on the right side as the radius. Always take the square root of that constant to find $r$. If the equation ends in $= 49$, the radius is $7$, not $49$.
• Coefficient Check: Before completing the square, ensure the coefficients of $x^2$ and $y^2$ are both $1$. If they are not (e.g., $2x^2 + 2y^2...$), divide the entire equation by that coefficient first.
• Verification: To check if a point lies on a circle, substitute its coordinates into the equation. If the left side equals the right side, the point is on the circumference.