Finding the Centre & Radius

The Standard Form of a circle's equation is defined as $(x - a)^2 + (y - b)^2 = r^2$. In this representation, the point $(a, b)$ represents the centre of the circle on a Cartesian plane, and $r$ represents the radius, which is the constant distance from the centre to any point on the circumference.

The General Form of a circle's equation is often expressed as $x^2 + y^2 + 2fx + 2gy + c = 0$. While this form is useful for general algebraic manipulation, it does not explicitly show the geometric properties of the circle, requiring conversion to the standard form for interpretation.

A circle is only valid in the real plane if the value of $r^2$ is strictly positive. If the algebraic manipulation results in $r^2 = 0$, the equation represents a single point; if $r^2 < 0$, the equation represents an imaginary circle with no real points.

The standard equation of a circle is derived directly from the Pythagorean Theorem or the distance formula. For any point $(x, y)$ on the circle, the distance to the centre $(a, b)$ must equal $r$, leading to the relation $\sqrt{(x-a)^2 + (y-b)^2} = r$.

Squaring both sides of the distance formula yields the standard form $(x - a)^2 + (y - b)^2 = r^2$. This principle ensures that every point satisfying the equation is exactly $r$ units away from the centre.

When an equation is given in general form, it represents an expanded version of the binomial squares. Reversing this expansion requires the algebraic technique of completing the square for both the $x$ and $y$ variables independently.

Step-by-Step Conversion

• Step 1: Grouping: Arrange the equation so that all $x$ terms are together and all $y$ terms are together, moving the constant term to the right-hand side of the equals sign.
• Step 2: Completing the Square: For the $x$ terms $x^2 + Bx$, add $(\frac{B}{2})^2$ to both sides of the equation. Repeat this process for the $y$ terms $y^2 + Dy$ by adding $(\frac{D}{2})^2$ to both sides.
• Step 3: Factoring: Rewrite the grouped $x$ and $y$ terms as perfect square binomials, such as $(x + \frac{B}{2})^2$ and $(y + \frac{D}{2})^2$.
• Step 4: Simplification: Combine all constants on the right-hand side to find the value of $r^2$.

Formulaic Shortcut

• For the general form $x^2 + y^2 + 2fx + 2gy + c = 0$, the centre is located at $(-f, -g)$. This is found by taking half the coefficients of $x$ and $y$ and changing their signs.
• The radius is calculated using the formula $r = \sqrt{f^2 + g^2 - c}$. This formula is derived from the constants produced during the completing the square process.

• Sign Interpretation: In the standard form $(x-a)^2$, a negative sign in the bracket indicates a positive coordinate for the centre, while a positive sign $(x+a)^2$ indicates a negative coordinate. This is a frequent source of error for students.

• Radius vs. Radius Squared: The right-hand side of the standard form is $r^2$, not $r$. One must always take the square root of the constant on the right to find the actual radius length.

• Coefficient Check: Before completing the square, ensure the coefficients of $x^2$ and $y^2$ are both equal to $1$. If they are not (e.g., $2x^2 + 2y^2...$), divide the entire equation by that coefficient first.

• The 'Half and Square' Rule: When completing the square, always remember to add the squared term to both sides of the equation. Forgetting to add it to the right-hand side will result in an incorrect radius.

• Sanity Check: If you calculate a radius and get a negative number under the square root, re-check your constant movements. A real circle must have a positive value for $f^2 + g^2 - c$.

• Sign Reversal: Always double-check the signs of the centre coordinates. If the equation is $(x+5)^2$, the x-coordinate of the centre is $-5$.