Line and Circle

A straight road runs through flat farmland. In the middle of a field, there is a circular pond of radius 50 metres. A surveyor standing at the road's edge asks: Does the road cross the pond, graze its boundary, or miss it entirely? The answer depends on one measurement — how far the road is from the centre of the pond.

If the shortest distance from the centre to the road is more than 50 metres, the road misses the pond. If it is exactly 50 metres, the road just touches the edge at one point. If it is less than 50 metres, the road cuts through the pond — entering at one point and leaving at another, creating a chord whose length you can compute.

That is the whole story of a line meeting a circle, and the algebra behind it is both clean and powerful.

The three cases

Consider a circle with centre C and radius r, and a straight line \ell. Drop a perpendicular from C to \ell. Call the foot of the perpendicular M, and call the perpendicular distance d = CM.

There are exactly three possibilities:

Case	Condition	Intersection points	Name
Line misses the circle	d > r	0	No intersection
Line touches the circle	d = r	1	Tangent
Line cuts through the circle	d < r	2	Secant

This is not a guess — it follows from the Pythagorean theorem. If the line intersects the circle at a point P, then triangle CMP is a right triangle with hypotenuse CP = r and one leg CM = d. The other leg MP satisfies MP = \sqrt{r^2 - d^2}. For real intersection points, you need r^2 - d^2 \geq 0, i.e., d \leq r.

When d < r, there are two intersection points — one on each side of M along the line, each at distance \sqrt{r^2 - d^2} from M. When d = r, the two points coincide at M — there is only one intersection point. When d > r, there are no real solutions.

The algebraic approach: substitution

The geometric picture above is elegant, but exams often require the algebraic method. Here is how it works.

Take a circle x^2 + y^2 = a^2 (centred at the origin, radius a) and a line y = mx + c. Substitute the line's equation into the circle's:

Why: substituting y = mx + c into x^2 + y^2 = a^2 eliminates y, leaving a quadratic in x. The number of real solutions of this quadratic tells you how many intersection points exist.

This is a quadratic equation in x. Its discriminant is

Why: expanding and simplifying the discriminant. The m^2c^2 terms cancel, leaving a clean expression involving a, m, and c.

The three cases now match the sign of the discriminant:

• D > 0 (i.e., c^2 < a^2(1 + m^2)): two intersection points — the line is a secant.
• D = 0 (i.e., c^2 = a^2(1 + m^2)): one intersection point — the line is tangent.
• D < 0 (i.e., c^2 > a^2(1 + m^2)): no intersection — the line misses the circle.

Connecting the two approaches

The perpendicular distance from the origin (0, 0) to the line y = mx + c, or equivalently mx - y + c = 0, is

The tangency condition D = 0 says c^2 = a^2(1 + m^2), which is \dfrac{c^2}{1 + m^2} = a^2, which is d^2 = a^2, i.e., d = a = r. The perpendicular distance equals the radius — exactly the geometric condition for tangency. The algebra and the geometry say the same thing.

Length of chord

When the line is a secant (d < r), it cuts a chord across the circle. The length of this chord has a clean formula.

Derivation

Let the perpendicular from the centre C meet the chord at M, and let the chord endpoints be A and B. Triangle CMA is right-angled at M, with CA = r (radius) and CM = d (perpendicular distance).

By the Pythagorean theorem:

The perpendicular from the centre to a chord bisects the chord (this is a standard circle theorem — the centre, the midpoint of the chord, and the foot of the perpendicular are all the same point). So MB = MA, and the full chord length is

Sanity check: when d = 0 (the line passes through the centre), the chord is a diameter with length 2r. When d = r (the line is tangent), the chord length is 0 — the "chord" is a single point of tangency. Both match.

The condition for tangency

The tangency condition deserves its own statement because it appears in so many problems.

This condition says: for a given slope m, there are exactly two tangent lines to the circle — one with c = +a\sqrt{1 + m^2} and one with c = -a\sqrt{1 + m^2}. They are parallel to each other, touching the circle on opposite sides.

For a general circle (x - h)^2 + (y - k)^2 = r^2, the condition for the line y = mx + c to be tangent is

which is just the statement that the perpendicular distance from the centre (h, k) to the line mx - y + c = 0 equals the radius r.

Equation of the tangent

There are several forms, each useful in different situations.

Form 1: Tangent at a point on the circle (point form)

Let P = (x_1, y_1) be a point on the circle x^2 + y^2 = a^2. You want the equation of the tangent at P.

Derivation. The radius OP has slope \dfrac{y_1}{x_1} (assuming x_1 \neq 0). The tangent at P is perpendicular to the radius — this is a fundamental property of circles. So the tangent has slope -\dfrac{x_1}{y_1}.

Using the point-slope form with point (x_1, y_1):

Multiply both sides by y_1:

Since (x_1, y_1) is on the circle, x_1^2 + y_1^2 = a^2. So

Why: the tangent is perpendicular to the radius at the point of contact. This is the single most important geometric property of a tangent to a circle. The algebraic derivation just translates that perpendicularity into an equation.

Notice the pattern: you get the tangent equation from the circle equation by replacing x^2 with xx_1 and y^2 with yy_1. This "replace and halve" rule (sometimes called the T = 0 substitution) extends to the general form.

For the general circle x^2 + y^2 + 2gx + 2fy + c = 0, the tangent at (x_1, y_1) is

The rule is: x^2 \to xx_1, y^2 \to yy_1, 2gx \to g(x + x_1), 2fy \to f(y + y_1), and c stays c.

Form 2: Tangent with a given slope (slope form)

Sometimes you know the slope but not the point of contact. You want the tangent to x^2 + y^2 = a^2 with slope m.

The tangent is y = mx + c where c satisfies the tangency condition c^2 = a^2(1 + m^2). So c = \pm a\sqrt{1 + m^2}, and the two tangent lines are

The \pm gives two lines — one on each side of the circle, both with slope m, both tangent.

Form 3: Tangent using the parametric point

Using the parametric form of the circle, any point on x^2 + y^2 = a^2 can be written as (a\cos\theta, a\sin\theta). Substituting x_1 = a\cos\theta and y_1 = a\sin\theta into the point form xx_1 + yy_1 = a^2:

This is the most compact form. It is useful when problems specify the point of tangency by its parameter \theta rather than by explicit coordinates.

Worked examples

Common confusions

• "A tangent touches the circle at two coincident points." This is a correct formal statement (the discriminant is zero, giving a repeated root), but it confuses many students. Geometrically, there is one point of contact. Algebraically, the quadratic has a repeated root, meaning two roots that happen to be equal. Both descriptions are valid — they are two ways of saying the same thing.

• "The tangent at a point is perpendicular to the circle." The tangent is perpendicular to the radius at the point of tangency, not to the circle itself. Saying "perpendicular to the circle" is imprecise — a circle is a curve, not a line, and "perpendicular to a curve at a point" means perpendicular to the tangent, which would give you the normal.

• "The condition c^2 = a^2(1 + m^2) works for any circle." This specific form works only for circles centred at the origin. For a general circle (x - h)^2 + (y - k)^2 = r^2, the condition is (mh - k + c)^2 = r^2(1 + m^2), which reduces to the simpler form only when h = k = 0.

• "In the point form xx_1 + yy_1 = a^2, I can use any point (x_1, y_1)." The point must be on the circle. If (x_1, y_1) is not on the circle, the equation xx_1 + yy_1 = a^2 still defines a line, but it is no longer tangent — it is the chord of contact if the point is external, or has no geometric meaning if the point is internal. The distinction matters.

• "Chord length can be negative." Never. The chord length is 2\sqrt{r^2 - d^2}, and since d \leq r for a secant, the expression under the root is non-negative. If you get a negative value under the root, the line does not intersect the circle — re-check whether you are dealing with a secant at all.

Going deeper

Tangent from an external point

An important class of problems: given a point P outside the circle, find the equations of the tangent lines from P to the circle.

Let the circle be x^2 + y^2 = a^2 and let P = (x_0, y_0) with x_0^2 + y_0^2 > a^2 (outside the circle). Any line through P with slope m is y - y_0 = m(x - x_0), or y = mx + (y_0 - mx_0).

For this line to be tangent, the tangency condition gives

Expanding:

This is a quadratic in m. It has two solutions (since P is outside the circle, the discriminant is positive), giving the slopes of the two tangent lines from P.

The length of each tangent — the distance from P to the point where the tangent touches the circle — is

Why: in the right triangle formed by the centre O, the point of tangency T, and the external point P, you have OP^2 = OT^2 + PT^2. So PT^2 = OP^2 - OT^2 = (x_0^2 + y_0^2) - a^2.

This formula is clean and worth remembering: the tangent length from an external point equals the square root of the power of the point with respect to the circle.

The normal at a point on the circle

The normal at a point on the circle is the line through that point perpendicular to the tangent. Since the tangent is perpendicular to the radius, the normal is along the radius — it passes through the centre.

For the circle x^2 + y^2 = a^2, the normal at (x_1, y_1) passes through the origin and through (x_1, y_1), so its equation is

or equivalently, xy_1 - yx_1 = 0.

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

The normal always passes through the centre. This is unique to circles — for other curves (ellipses, parabolas), the normal generally does not pass through any fixed point.

Connection to quadratic equations

The intersection of a line and a circle always reduces to solving a quadratic equation. The three cases of the discriminant — positive, zero, negative — correspond exactly to the three geometric configurations: secant, tangent, no intersection.

This is not a coincidence. It is a manifestation of a deeper principle: whenever a straight line (degree 1) meets a conic section (degree 2), the intersection is governed by an equation of degree 1 + 2 - 1 = 2, i.e., a quadratic. The quadratic can have two real roots (secant), one repeated root (tangent), or two complex roots (no real intersection). This pattern persists for all conics — parabolas, ellipses, hyperbolas — not just circles.

The director circle

Here is an elegant result that connects tangent lines to a new circle. The director circle of a given circle is the locus of the point from which the two tangent lines to the given circle are perpendicular to each other.

For the circle x^2 + y^2 = a^2, if the two tangent lines from a point P(x_0, y_0) are perpendicular, then the product of their slopes is -1. From the quadratic in m derived earlier, the product of the roots is \dfrac{y_0^2 - a^2}{x_0^2 - a^2}. Setting this equal to -1:

The locus is the circle x^2 + y^2 = 2a^2 — a circle centred at the same point, with radius a\sqrt{2}. The director circle is concentric with the original, and its radius is \sqrt{2} times larger.

Where this leads next

The interaction between lines and circles is the foundation for a large body of results. The most direct continuations:

• Tangent and Normal to Circle — the full treatment of tangent and normal lines, including the length of the tangent from an external point and the angle between tangent and radius.
• Pair of Tangents and Chord of Contact — the two tangent lines from an external point, the chord joining the contact points, and the equation of the chord of contact.
• Circle — Standard Forms — the foundation article on the two forms of the circle equation, if you want to review.
• Distance Formulas — the perpendicular-distance formula that drives the tangency condition.
• Quadratic Equations — the discriminant theory that governs the three cases of line-circle intersection.