Finding the Centre & Radius

Core Definition: The standard equation of a circle is given by $(x - a)^2 + (y - b)^2 = r^2$. This form is derived from the distance formula, representing all points $(x, y)$ that are exactly $r$ units away from a fixed point $(a, b)$.

Identifying Components: In this format, the coordinates of the centre are $(a, b)$ and the radius is $r$. It is crucial to note that the signs inside the brackets are negative, meaning the centre coordinates are the opposites of the numbers appearing with $x$ and $y$.

The Radius Squared: The constant on the right-hand side of the equation represents $r^2$, not the radius itself. To find the actual radius, one must always take the positive square root of this constant value.

Structure: Circles are often presented in the expanded general form: $x^2 + y^2 + 2fx + 2gy + c = 0$. This form is less intuitive because the centre and radius are hidden within the coefficients of the linear terms and the constant.

Coefficient Relationship: In this specific notation, the centre of the circle is located at $(-f, -g)$. This means you must divide the coefficients of $x$ and $y$ by 2 and then change their signs to find the centre coordinates.

Radius Formula: The radius can be calculated directly from the general form using the relationship $r = \sqrt{f^2 + g^2 - c}$. For a valid circle to exist, the value under the square root must be strictly positive.

Step 1: Grouping: Begin by rearranging the equation to group the $x$-terms together and the $y$-terms together, while moving the constant term to the right side of the equals sign.

Step 2: Creating Squares: For both the $x$ and $y$ groups, add the square of half the coefficient of the linear term. To maintain equality, you must add these same values to the right side of the equation.

Step 3: Factoring: Rewrite the grouped terms as perfect square binomials, such as $(x - a)^2$ and $(y - b)^2$. The right side will now simplify to a single numerical value representing $r^2$.

The Sign Flip: Always double-check the signs when extracting the centre. A common exam trap is providing $(x + 5)^2$, where the x-coordinate is $-5$, not $+5$.

Coefficient Check: Before completing the square, ensure the coefficients of $x^2$ and $y^2$ are both exactly $1$. If they are not, you must divide the entire equation by that coefficient first.

Sanity Check for Radius: If your calculation for $r^2$ results in a negative number, re-examine your algebra. A negative $r^2$ implies the equation does not represent a real circle in the Cartesian plane.