Tangent and Normal to Circle

Take the circle x^2 + y^2 = 25 — centre at the origin, radius 5. Pick the point (3, 4) on the circle (check: 9 + 16 = 25). Now try to draw a line through (3, 4) that meets the circle at exactly one point — no second crossing, no cutting through. The line should just graze the circle and leave.

There is exactly one such line, and it has a striking property: it is perpendicular to the radius drawn from the origin to (3, 4). The radius points in the direction (3, 4), with slope 4/3. The line perpendicular to it must have slope -3/4. So the tangent line is

That is a tangent to the circle at the point (3, 4).

You can picture the tangent as the wall against which the circle rests. If you placed a circular coin on a table and pushed a ruler against its edge, the ruler would be the tangent — flat against the coin at exactly one point, and perpendicular to the line from the coin's centre to that contact point. Temple architects in India have used this relationship for centuries: the stone floor is tangent to the base of each pillar, and the load transmits along the normal — straight through the centre.

Now here is the deeper question: why is the tangent perpendicular to the radius? And can you write down the tangent at any point on any circle without going through slope calculations each time? The answer to both questions leads to clean, memorable formulas that you will use constantly in coordinate geometry.

Why the tangent is perpendicular to the radius

Start with the circle x^2 + y^2 = r^2 and a point P(x_1, y_1) on it, so x_1^2 + y_1^2 = r^2. Suppose a line through P meets the circle at another point Q. The centre is the origin O.

The midpoint of the chord PQ lies on the perpendicular from the centre to the chord — that is a theorem you know from circle geometry. As Q slides along the circle toward P, the chord PQ gets shorter. The midpoint of the chord moves toward P. At the instant Q coincides with P, the chord has collapsed to a single point, and the "perpendicular from the centre to the chord" has become the radius OP itself.

The line that was the chord PQ has, in the limit, become the tangent. And the perpendicular from the centre to that line is the radius. So the tangent is perpendicular to the radius.

This is not a coincidence — it is the geometric essence of what "tangent" means. A tangent is a chord whose two endpoints have merged into one. The perpendicularity is built into the definition.

Equation of tangent at a point on the circle

For x^2 + y^2 = r^2

Let P(x_1, y_1) lie on x^2 + y^2 = r^2. The slope of OP is y_1/x_1. The tangent at P is perpendicular to OP, so its slope is -x_1/y_1 (provided y_1 \neq 0). Using the point-slope form:

Multiply both sides by y_1:

Look at how clean this is. The equation of the circle is x^2 + y^2 = r^2. The equation of the tangent at (x_1, y_1) is xx_1 + yy_1 = r^2. You replace each squared variable with the product of the variable and its value at the point. This pattern — called the T-substitution — works for every conic, not just circles. You will see it again for parabolas, ellipses, and hyperbolas.

What if y_1 = 0? Then the point is (r, 0) or (-r, 0), and the tangent is x = r or x = -r — a vertical line. The formula xx_1 + yy_1 = r^2 gives xr = r^2, i.e. x = r. It still works.

Try it at a few points on the circle x^2 + y^2 = 25:

For the general circle

For the general circle x^2 + y^2 + 2gx + 2fy + c = 0, the same T-substitution applies: replace x^2 with xx_1, y^2 with yy_1, 2x with (x + x_1), and 2y with (y + y_1).

This looks more complex, but it is the same pattern. When you write the circle in the form (x + g)^2 + (y + f)^2 = g^2 + f^2 - c, the centre is (-g, -f) and the radius is \sqrt{g^2 + f^2 - c}. The derivation follows the same perpendicularity argument: the tangent is perpendicular to the line joining (-g, -f) to (x_1, y_1).

Tangent with a given slope

Sometimes you do not know the point of contact — you know the slope the tangent should have, and you want to find which tangent line(s) of that slope exist.

Take x^2 + y^2 = r^2 and a line y = mx + c. From the article on Line and Circle, this line is tangent to the circle when the discriminant of the resulting quadratic is zero — equivalently, when the perpendicular distance from the centre to the line equals the radius.

The distance from (0, 0) to the line mx - y + c = 0 is \dfrac{|c|}{\sqrt{m^2 + 1}}. Setting this equal to r:

The "\pm" carries geometric meaning: the two tangent lines with the same slope sit on opposite sides of the centre, like a pair of parallel rails with the circle squeezed between them.

For the general circle x^2 + y^2 + 2gx + 2fy + c = 0 with centre (-g, -f) and radius R = \sqrt{g^2 + f^2 - c}, the tangent of slope m is

This is the same distance condition — the perpendicular distance from the centre (-g, -f) to y = mx + k must equal R.

Length of tangent from an external point

Here is one of the most satisfying results in circle geometry. Take a point P(x_1, y_1) outside the circle x^2 + y^2 + 2gx + 2fy + c = 0. Draw a tangent from P to the circle, touching it at T. The segment PT has a specific length, and you can compute it without knowing T.

The centre of the circle is C(-g, -f) and the radius is R. Since PT is tangent to the circle at T, the angle \angle PTC = 90°. The triangle PTC is a right triangle with hypotenuse PC.

By the Pythagorean theorem:

Now compute PC^2:

And R^2 = g^2 + f^2 - c. So:

That expression on the right is exactly what you get when you substitute (x_1, y_1) into the left side of the circle equation x^2 + y^2 + 2gx + 2fy + c. This substituted value is conventionally called S_1.

This is worth pausing on. The "value of the circle equation at a point" has direct geometric meaning:

• If S_1 = 0, the point is on the circle (tangent length is zero — the point is the point of tangency).
• If S_1 > 0, the point is outside the circle, and \sqrt{S_1} is the tangent length.
• If S_1 < 0, the point is inside the circle, and no real tangent exists.

The sign of S_1 tells you the position of the point relative to the circle. This idea will return powerfully in the articles on chord of contact and radical axis.

Equal tangent lengths

From any external point, two tangent lines can be drawn to a circle. The two tangent segments — from the external point to the respective points of contact — always have the same length. This follows immediately from the formula: both tangent lengths equal \sqrt{S_1}, which depends only on the external point, not on which tangent you pick.

This symmetry has a nice geometric reading. The two tangent lines from P to a circle are mirror images of each other, reflected across the line joining P to the centre. The two triangles formed — PTC for each tangent point T — are congruent right triangles (same hypotenuse PC, same leg R), so the third side PT must be equal.

Normal to a circle

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

This is a simple but important fact: the normal to a circle at any point passes through the centre.

For the circle x^2 + y^2 = r^2 and the point P(x_1, y_1), the normal is the line joining the origin to P:

or equivalently, xy_1 - yx_1 = 0.

For the general circle with centre (-g, -f), the normal at (x_1, y_1) is the line through (x_1, y_1) and (-g, -f):

Worked examples

Common confusions

• "The tangent touches the circle, so it intersects at zero points." No — it intersects at exactly one point. A line that misses the circle entirely intersects at zero points. The tangent is the boundary case: it meets the circle, but only barely.

• "The formula xx_1 + yy_1 = r^2 works for any point." Only for points on the circle. If (x_1, y_1) is not on x^2 + y^2 = r^2, the line xx_1 + yy_1 = r^2 is still a well-defined line, but it is not a tangent — it is the chord of contact, which you will meet in the next article.

• "There is only one tangent from an external point." There are always two. From any external point, you can draw exactly two tangent lines to a circle. Both tangent segments have the same length \sqrt{S_1} — this is a consequence of the symmetry of the circle.

• "The normal to a curve always passes through the centre." Only for circles. The normal to a parabola, ellipse, or hyperbola does not pass through the centre (or focus). The circle is special because its curvature is constant — every point is at the same distance from the centre, so the radius direction and the normal direction are always the same.

Going deeper

The calculus connection

If you have met derivatives, there is a second way to find the tangent. Take the circle x^2 + y^2 = r^2. Differentiate both sides implicitly with respect to x:

At (x_1, y_1), the slope of the tangent is -x_1/y_1. The tangent line is:

Multiply through by y_1 and simplify: yy_1 - y_1^2 = -xx_1 + x_1^2, which gives xx_1 + yy_1 = x_1^2 + y_1^2 = r^2.

This is exactly the same formula you derived geometrically. The calculus derivation confirms it: the T-substitution is not a trick — it is what implicit differentiation produces when applied to the equation of a circle.

Why the condition c^2 = r^2(1 + m^2) works

For the line y = mx + c to be tangent to x^2 + y^2 = r^2, substitute into the circle equation:

This is a quadratic in x. The line is tangent when the quadratic has exactly one solution — that is, discriminant equals zero:

The tangency condition is the discriminant condition in disguise. A tangent is a line that meets the curve at a "double point" — two intersections that have collapsed into one.

Parametric form of the tangent

The circle x^2 + y^2 = r^2 has a natural parametrisation: every point on it can be written as (r\cos\theta, r\sin\theta) for some angle \theta. The tangent at the point corresponding to angle \theta is obtained by substituting x_1 = r\cos\theta, y_1 = r\sin\theta into xx_1 + yy_1 = r^2:

This is the normal form of a line at distance r from the origin, making angle \theta with the positive x-axis. The tangent to a circle at angle \theta is exactly the line whose perpendicular from the origin has length r and direction \theta. This connects the tangent formula to the normal form of a line from Straight Line — Forms.

Director circle

From an external point P, the two tangents to the circle make a certain angle between them. There is a special set of points for which this angle is exactly 90° — the two tangent lines are perpendicular. The locus of all such points is itself a circle, called the director circle.

For x^2 + y^2 = r^2, the director circle is x^2 + y^2 = 2r^2. Its radius is r\sqrt{2} — exactly \sqrt{2} times the original radius.

To derive this, let P(h, k) be a point from which both tangents to the circle have perpendicular slopes m_1 and m_2 with m_1 m_2 = -1. The tangent from P with slope m must satisfy k = mh \pm r\sqrt{1 + m^2}, i.e. (k - mh)^2 = r^2(1 + m^2). Expand: k^2 - 2mkh + m^2 h^2 = r^2 + r^2 m^2, which gives (h^2 - r^2)m^2 - 2khm + (k^2 - r^2) = 0. The product of the roots is m_1 m_2 = \dfrac{k^2 - r^2}{h^2 - r^2}. Setting this to -1: k^2 - r^2 = -(h^2 - r^2), so h^2 + k^2 = 2r^2. The locus is indeed x^2 + y^2 = 2r^2.

Where this leads next

You now have the tools to find tangents and normals at individual points and with given slopes. The natural extensions:

• Pair of Tangents and Chord of Contact — when you draw both tangents from an external point, the line joining their points of contact has a name and a formula.
• Family of Circles — circles passing through the intersection of two given circles, the radical axis, and orthogonal circles.
• Line and Circle — the prerequisite to this article, covering when a line cuts, touches, or misses a circle.
• Circle — Standard Forms — the equations of circles in all standard forms.
• Derivative — the calculus concept that gives tangent lines their formal definition for arbitrary curves.