8.3 Circle Equations, Tangents, Normals & Intersections

Circle Equations, Tangents, Normals & Intersections

In analytic geometry and ECAT Mathematics, the circle represents the most fundamental non-degenerate quadratic locus. Taught in Intermediate Mathematics Part II, circle questions test a candidate's ability to rapidly compute centers, radii, tangent conditions, and intersection metrics without tedious calculus. Mastering the algebraic notation of circles ($S = 0, S_1, T = 0$) allows students to solve multi-step problems in under 45 seconds.

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1. Geometric Definition and Standard Circle Equations

A circle is the locus of a planar point $P(x, y)$ that maintains a constant positive Euclidean distance $r$ (the radius) from a fixed reference point $C(h, k)$ (the center).

Standard (Center-Radius) Form

Applying the distance formula $|CP| = r$ yields the standard equation of a circle:

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

For a circle centered at the origin $(0, 0)$ with radius $r$:

$x^2 + y^2 = r^2$

Parametric Form

Every point on $x^2 + y^2 = r^2$ can be parameterized in terms of the polar angle $\theta \in [0, 2\pi)$:

$x = r\cos\theta, \quad y = r\sin\theta$

For the shifted circle $(x - h)^2 + (y - k)^2 = r^2$, the parametric representation is:

$x = h + r\cos\theta, \quad y = k + r\sin\theta$

Diametric Form

If the endpoints of a diameter are given as $A(x_1, y_1)$ and $B(x_2, y_2)$, any point $P(x, y)$ on the circumference subtends a right angle ($90^\circ$) at $A$ and $B$. Consequently, the product of the slopes of segments $AP$ and $BP$ equals $-1$, yielding the diametric equation:

$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$

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2. General Equation of a Second Degree Representing a Circle

The general second-degree equation in two variables is:

$a x^2 + 2h xy + b y^2 + 2g x + 2f y + c = 0$

This equation represents a circle if and only if two structural criteria are fulfilled:

1. Equality of Quadratic Coefficients: $a = b \neq 0$.
2. Absence of the Product Term: $h = 0$ (no $xy$ term).

Dividing throughout by $a$ yields the standard monic general form:

$x^2 + y^2 + 2gx + 2fy + c = 0$

Completing the Square for Center and Radius

Grouping terms and completing squares:

$(x^2 + 2gx + g^2) + (y^2 + 2fy + f^2) = g^2 + f^2 - c \implies (x + g)^2 + (y + f)^2 = g^2 + f^2 - c$

Matching this with $(x - h)^2 + (y - k)^2 = r^2$ establishes:

$\text{Center: } C(-g, -f) = \left( -\frac{\text{Coeff of } x}{2}, \; -\frac{\text{Coeff of } y}{2} \right)$

$\text{Radius: } r = \sqrt{g^2 + f^2 - c}$

Nature of the Circle

The geometric nature of the locus is governed by the discriminant $g^2 + f^2 - c$:

• Real Circle: If $g^2 + f^2 - c > 0$, the radius is a positive real number, yielding an ordinary circle.
• Point Circle: If $g^2 + f^2 - c = 0$, the radius is zero; the equation represents the single isolated real point $(-g, -f)$.
• Imaginary (Virtual) Circle: If $g^2 + f^2 - c < 0$, the radius is imaginary; no real point $(x, y) \in \mathbb{R}^2$ satisfies the equation.

ECAT Trap Alert: Before extracting $g, f,$ and $c$, always divide the entire equation by the coefficient of $x^2$ (or $y^2$). For example, in $2x^2 + 2y^2 - 8x + 12y - 6 = 0$, divide by 2 first: $x^2 + y^2 - 4x + 6y - 3 = 0$. Here $g = -2, f = 3, c = -3$, giving center $(2, -3)$ and $r = \sqrt{(-2)^2 + 3^2 - (-3)} = \sqrt{4 + 9 + 3} = \sqrt{16} = 4$.

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3. Position of a Point Relative to a Circle & Tangent Length

Let the general circle equation be denoted by:

$S \equiv x^2 + y^2 + 2gx + 2fy + c = 0$

For any point $P(x_1, y_1)$ in the Cartesian plane, define the power scalar $S_1$ by substituting $(x_1, y_1)$ into the monic expression:

$S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$

Geometrically, $S_1 = |CP|^2 - r^2$, leading to three distinct positional states:

1. $S_1 > 0$: Point $P$ lies strictly outside the circle ($|CP| > r$). Two real tangents can be drawn from $P$ to the circle.
2. $S_1 = 0$: Point $P$ lies on the circumference ($|CP| = r$). Exactly one real tangent touches the circle at $P$.
3. $S_1 < 0$: Point $P$ lies strictly inside the circle ($|CP| < r$). No real tangents can be drawn from $P$.

Length of Tangent from an External Point

For a point $P(x_1, y_1)$ with $S_1 > 0$, the straight-line segment from $P$ to either point of tangency $T$ forms a right-angled triangle $\triangle CPT$ with hypotenuse $|CP|$ and leg $|CT| = r$. By the Pythagorean theorem, the length of the tangent $L$ is:

$L = |PT| = \sqrt{|CP|^2 - r^2} = \sqrt{S_1} = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$

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4. Tangent and Normal Lines to a Circle

The Standard Tangent Transformation ($T = 0$)

The equation of the tangent at point $(x_1, y_1)$ lying on circle $S = 0$ is obtained via the universal quadratic substitution:

$x^2 \to xx_1, \quad y^2 \to yy_1, \quad 2x \to x + x_1, \quad 2y \to y + y_1, \quad xy \to \frac{xy_1 + x_1 y}{2}$

Defining the linear operator $T$:

$T \equiv xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$

For the central circle $x^2 + y^2 = r^2$, the tangent at $(x_1, y_1)$ is simply:

$xx_1 + yy_1 = r^2$

In parametric coordinates $(r\cos\theta, r\sin\theta)$:

$x\cos\theta + y\sin\theta = r$

Condition of Tangency for a Line

Consider the straight line $y = mx + c$ and the circle $x^2 + y^2 = r^2$. The line is tangent to the circle if and only if the perpendicular distance from the center $(0, 0)$ to the line equals the radius $r$:

$\frac{|c|}{\sqrt{1 + m^2}} = r \implies c = \pm r\sqrt{1 + m^2}$

Thus, the equations of tangents with slope $m$ are:

$y = mx \pm r\sqrt{1 + m^2}$

The coordinates of the points of contact are:

$\left( \mp \frac{m r}{\sqrt{1 + m^2}}, \; \pm \frac{r}{\sqrt{1 + m^2}} \right)$

Equation of the Normal Line

The normal to a circle at any boundary point $P(x_1, y_1)$ is the straight line perpendicular to the tangent at $P$. Because the radius is perpendicular to the tangent at the point of contact, every normal line to a circle passes through its center $(-g, -f)$.

• For circle $x^2 + y^2 = r^2$, the normal at $(x_1, y_1)$ connects the origin $(0, 0)$ and $(x_1, y_1)$:

$\frac{x}{x_1} = \frac{y}{y_1} \iff x y_1 - y x_1 = 0$

• For the general circle $x^2 + y^2 + 2gx + 2fy + c = 0$, the normal passes through $(-g, -f)$ and $(x_1, y_1)$:

$(y - y_1)(x_1 + g) = (x - x_1)(y_1 + f)$

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5. Chords of Contact and the Director Circle

Chord of Contact

If two tangents are drawn from an external point $P(x_1, y_1)$ touching the circle at $T_1$ and $T_2$, the straight line joining $T_1$ and $T_2$ is termed the chord of contact. Its equation is given by the same linear operator:

$T = 0 \implies xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$

ECAT Distinction:

• If $(x_1, y_1)$ is on the circle, $T = 0$ is the tangent line.
• If $(x_1, y_1)$ is outside the circle, $T = 0$ is the chord of contact.
• If $(x_1, y_1)$ is inside the circle, $T = 0$ represents the polar line of $P$.

Chord with a Given Midpoint

The equation of a chord of circle $S = 0$ whose midpoint is known to be $(x_1, y_1)$ is:

$T = S_1$

The Director Circle

The director circle is defined as the locus of the point of intersection of two mutually perpendicular tangents drawn to a circle.

Let the tangents to $x^2 + y^2 = r^2$ be $y - mx = \pm r\sqrt{1 + m^2}$. Squaring and rearranging:

$(y - mx)^2 = r^2(1 + m^2) \implies (x^2 - r^2)m^2 - 2xy m + (y^2 - r^2) = 0$

For the two tangents to be perpendicular, the product of their slopes must be $-1$:

$m_1 m_2 = \frac{y^2 - r^2}{x^2 - r^2} = -1 \implies y^2 - r^2 = -(x^2 - r^2) \implies x^2 + y^2 = 2r^2$

Hence, the director circle of $x^2 + y^2 = r^2$ is a concentric circle with radius $r\sqrt{2}$.

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6. Radical Axis and Orthogonal Intersections

Radical Axis and Common Chord

For two non-concentric circles $S \equiv x^2 + y^2 + 2g_1 x + 2f_1 y + c_1 = 0$ and $S' \equiv x^2 + y^2 + 2g_2 x + 2f_2 y + c_2 = 0$, the radical axis is the locus of points from which the lengths of tangents to both circles are equal ($S = S'$):

$S - S' = 0 \implies 2(g_1 - g_2)x + 2(f_1 - f_2)y + (c_1 - c_2) = 0$

• The radical axis is always perpendicular to the line joining the centers $C_1 C_2$.
• If the circles intersect at two real points, the radical axis is their common chord.
• If the circles touch each other, the radical axis is their common tangent.

Orthogonal Circles

Two circles intersect orthogonally if the tangents at their points of intersection are perpendicular. In that configuration, the tangent to one circle passes through the center of the other, forming a right triangle with vertices $C_1, C_2,$ and the intersection point. By the Pythagorean theorem:

$|C_1 C_2|^2 = r_1^2 + r_2^2$

Expanding in terms of coordinates:

$(g_1 - g_2)^2 + (f_1 - f_2)^2 = (g_1^2 + f_1^2 - c_1) + (g_2^2 + f_2^2 - c_2)$

Cancelling quadratic terms yields the fundamental orthogonality condition:

$2g_1 g_2 + 2f_1 f_2 = c_1 + c_2$