Co-ordinate Geometry The Circle

Standard Form and General Form

A circle is defined as the set of all points $(x, y)$ that sit at a fixed radius length $r$ from a fixed centre point. When the centre is the origin $(0, 0)$, applying Pythagoras' theorem gives $x^2 + y^2 = r^2$. Learn this equation and understand where it comes from (the distance from the centre is always r). Check your own copy of the Formulae and Tables booklet to see which circle formulas it lists.

When the centre shifts to $(h, k)$, the distance formula gives the standard form $(x - h)^2 + (y - k)^2 = r^2$. Notice how the signs flip inside the brackets: an equation reading $(x - 5)^2 + (y + 3)^2 = 49$ has centre $(5, -3)$ and radius $r = 7$.

Expanding $(x - h)^2 + (y - k)^2 = r^2$ gives $x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0$. This expansion is usually written in the general form:

Comparing coefficients shows that $2g = -2h$ and $2f = -2k$, so the centre is $(-g, -f)$. The constant term is $c = g^2 + f^2 - r^2$, which rearranges to give the radius $r = \sqrt{g^2 + f^2 - c}$. Before extracting $g$, $f$, or $c$, always divide across by any common coefficient attached to $x^2$ and $y^2$ so that both squared terms start with $1$.

Testing Points and Axis Intercepts

To determine whether a point $(x_1, y_1)$ sits inside, outside, or on a circle, you compare its distance from the centre with the radius. When working with the standard form, substitute the point into $(x_1 - h)^2 + (y_1 - k)^2$. If the result is less than $r^2$, the point is inside; if it equals $r^2$, it lies on the boundary; if it exceeds $r^2$, it is outside.

If the circle is given in general form, substitute the point directly into the expression $x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$. A negative value means the point is inside, zero means on the circle, and a positive value means outside. For example, testing $(5, 4)$ in $x^2 + y^2 + 2x - 6y - 15 = 0$ gives $25 + 16 + 10 - 24 - 15 = 12 > 0$, confirming that $(5, 4)$ is outside.

To find where a circle crosses the coordinate axes, set the opposing coordinate to zero. Letting $y = 0$ leaves a quadratic in $x$ whose solutions are the $x$-intercepts. Similarly, setting $x = 0$ yields the $y$-intercepts. When a circle touches the $x$-axis, the radius equals $|-f|$, so $g^2 + f^2 - c = f^2$, which simplifies to the useful condition $g^2 = c$.

Lines and Circles: Intersections and Tangency

A line and a circle can cross at two distinct points, touch at one single point of contact as a tangent, or miss each other entirely. You can find exact intersection points by rearranging the linear equation for one variable, substituting that expression into the circle equation, solving the resulting quadratic equation in one variable, and substituting each solution back into the line equation to find the matching co-ordinate.

For example, to find where the line $x - y + 1 = 0$ meets the circle $x^2 + y^2 = 25$:

1. From the line, express $y$ in terms of $x$: $y = x + 1$.
2. Substitute into the circle equation: $x^2 + (x + 1)^2 = 25$, which gives $2x^2 + 2x - 24 = 0$, or $x^2 + x - 12 = 0$.
3. Factorise: $(x + 4)(x - 3) = 0$, so $x = 3$ or $x = -4$.
4. Substitute back into $y = x + 1$: $x = 3 \Rightarrow y = 4$, and $x = -4 \Rightarrow y = -3$. The points of intersection are $(3, 4)$ and $(-4, -3)$ (check: $3^2 + 4^2 = 25$).

The quadratic discriminant $b^2 - 4ac$ tells you the geometric arrangement without needing to find the coordinates. If $b^2 - 4ac > 0$, the line crosses the circle twice. If $b^2 - 4ac = 0$, the line is a tangent touching at one point. If $b^2 - 4ac < 0$, the line does not meet the circle.

When a problem asks you to prove tangency or find an unknown constant, using the perpendicular distance formula is almost always faster than algebra substitution. A straight line is tangent to a circle if and only if the perpendicular distance $d$ from the centre $(x_1, y_1)$ to the line $ax + by + c = 0$ equals the radius $r$:

Constructing Tangent Equations

Finding a tangent depends on whether the given point lies on the circle or outside it. By geometric theorem, every tangent line is perpendicular to the radius drawn to the point of contact.

If the point of contact $(x_1, y_1)$ is on the circle, first find the slope of the radius connecting the centre to that point using $m_r = \frac{y_1 - k}{x_1 - h}$. The tangent slope is its negative reciprocal, $m_t = -\frac{1}{m_r}$. Then write the line equation using $y - y_1 = m_t(x - x_1)$.

If the point $(x_1, y_1)$ lies outside the circle, write the tangent as an unknown line through that point: $y - y_1 = m(x - x_1)$, which rearranges to $mx - y + (y_1 - mx_1) = 0$. Apply the perpendicular distance formula from the centre to this line and set it equal to $r$. Squaring both sides produces a quadratic equation in $m$, yielding the two tangent slopes that pass through that external point.

Touching Circles and Tangent Lengths

Two circles touch each other when their boundaries meet at exactly one point. You test this by calculating the distance $d = |c_1c_2|$ between their centres and comparing it with their radii $r_1$ and $r_2$.

For external contact, the two circles sit outside each other and touch at a single point between their centres. This occurs when $d = r_1 + r_2$. For internal contact, one circle sits inside the other, which occurs when $d = |r_1 - r_2|$. In an exam, state both values explicitly and finish with a concluding sentence showing that they match.

Another classic exam question asks for the length of a tangent drawn from an external point $P$ to a circle with centre $C$ and radius $r$. Because the radius to the contact point $T$ meets the tangent at $90^\circ$, triangle $PTC$ is right-angled at $T$. Pythagoras' theorem gives $|PT|^2 + r^2 = |CP|^2$, so the tangent length is $|PT| = \sqrt{|CP|^2 - r^2}$.

Key terms

Standard Form

The circle equation $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the centre and $r$ is the radius.

General Form

The circle equation $x^2 + y^2 + 2gx + 2fy + c = 0$, with centre $(-g, -f)$ and radius $r = \sqrt{g^2 + f^2 - c}$.

Point of Contact

The single point where a tangent line touches a circle, forming a right angle with the radius drawn to that point.

Perpendicular Distance

The shortest distance from a point to a straight line; a line is tangent to a circle if and only if this distance from the centre equals the radius.

External Contact

The geometric condition where two separate circles touch at one boundary point, satisfying $|c_1c_2| = r_1 + r_2$.

Internal Contact

The geometric condition where one circle sits inside another and touches it at one point, satisfying $|c_1c_2| = |r_1 - r_2|$.