Circles: Tangents, Normals and Chord of Contact | NIMCET

Quick answer: For the circle $S: x^2 + y^2 + 2gx + 2fy + c = 0$ , the tangent at $(x_1, y_1)$ is $T = 0$ , where $T = xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c$ . The line $y = mx + c$ touches $x^2 + y^2 = a^2$ when $c^2 = a^2(1 + m^2)$ . The length of the tangent from an external point is $\sqrt{S_1}$ , and the chord of contact from that point is also $T = 0$ .

This continues from the equation of a circle.

Notation

Symbol	Meaning
$S$	$x^2 + y^2 + 2gx + 2fy + c$
$S_1$	S evaluated at $(x_1, y_1)$
$T$	$xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c$

For $x^2 + y^2 = a^2$ , $T$ is simply $xx_1 + yy_1 - a^2$ .

Tangents

Situation	Tangent
At $(x_1, y_1)$ on $x^2 + y^2 = a^2$	$xx_1 + yy_1 = a^2$
At $(x_1, y_1)$ on the general circle	$T = 0$
Slope m to $x^2 + y^2 = a^2$	$y = mx \pm a\sqrt{1 + m^2}$
Parametric point θ on $x^2 + y^2 = a^2$	$x\cos\theta + y\sin\theta = a$

Condition of tangency: $y = mx + c$ touches $x^2 + y^2 = a^2$ if

$c^2 = a^2(1 + m^2)$

Normal

The normal at any point passes through the centre. So the normal at $(x_1, y_1)$ is the line joining $(x_1, y_1)$ to $(-g, -f)$ .

Length of tangent and power of a point

From an external point $P(x_1, y_1)$ :

$\text{Length of tangent} = \sqrt{S_1}$

$S_1$ is called the power of the point. It is positive outside, zero on the circle, and negative inside.

Chord of contact and chord with a given midpoint

Chord	Equation
Chord of contact from $(x_1, y_1)$	$T = 0$
Chord with midpoint $(x_1, y_1)$	$T = S_1$

Two circles

Let the centres be $C_1, C_2$ , the radii $r_1, r_2$ , and $d = C_1C_2$ .

Condition	Position	Common tangents
$d > r_1 + r_2$	Separate	4
$d = r_1 + r_2$	Touch externally	3
$\lvert r_1 - r_2\rvert < d < r_1 + r_2$	Intersect	2
$d = \lvert r_1 - r_2\rvert$	Touch internally	1
$d < \lvert r_1 - r_2\rvert$	One inside the other	0

Orthogonal circles (tangents at an intersection point are perpendicular):

$2g_1g_2 + 2f_1f_2 = c_1 + c_2$

Family of circles

Family	Equation
Through the intersection of circle S and line L	$S + \lambda L = 0$
Through the intersection of two circles	$S_1 + \lambda S_2 = 0$ ( $\lambda \ne -1$ )
Common chord of two circles	$S_1 - S_2 = 0$

Worked examples

Example 1: tangent at a point

Find the tangent to $x^2 + y^2 = 25$ at $(3, 4)$ .

$3x + 4y = 25$ .

Example 2: tangency condition

For what c does $y = 2x + c$ touch $x^2 + y^2 = 5$ ?

$c^2 = 5(1 + 4) = 25$ , so $c = \pm 5$ .

Example 3: length of tangent

Find the tangent length from $(5, 1)$ to $x^2 + y^2 - 2x + 4y - 4 = 0$ .

$S_1 = 25 + 1 - 10 + 4 - 4 = 16$ , so the length is 4.

Example 4: orthogonal circles

Check whether $x^2 + y^2 - 2x - 4y + 1 = 0$ and $x^2 + y^2 + 4x - 2y + 3 = 0$ are orthogonal.

$g_1 = -1$ , $f_1 = -2$ , $c_1 = 1$ ; $g_2 = 2$ , $f_2 = -1$ , $c_2 = 3$ .

$2(-1)(2) + 2(-2)(-1) = -4 + 4 = 0$ , but $c_1 + c_2 = 4$ , so they are not orthogonal.

Changing the first circle to $x^2 + y^2 - 2x - 4y - 3 = 0$ gives $c_1 + c_2 = 0$ , and then they are orthogonal.

Summary table

Result	Formula
Tangent at a point	$T = 0$
Tangency condition	$c^2 = a^2(1 + m^2)$
Tangent length	$\sqrt{S_1}$
Chord with midpoint	$T = S_1$
Orthogonality	$2g_1g_2 + 2f_1f_2 = c_1 + c_2$

Exam tips

Make sure the coefficients of $x^2$ and $y^2$ are 1 before computing $S_1$ .
Count common tangents by comparing d with $r_1 + r_2$ and $\lvert r_1 - r_2\rvert$ .
The common chord is found by subtracting the two circle equations.

Next, study the parabola: focus, directrix and standard forms, or see the NIMCET Coordinate Geometry and Vectors chapter.

Common mistakes

Using $T = 0$ for the chord with a given midpoint; it is $T = S_1$ .
Forgetting the ± sign in $y = mx \pm a\sqrt{1 + m^2}$ .
Computing tangent length from a point inside the circle.
Using the orthogonality condition with unhalved coefficients.

Frequently Asked Questions

What is the equation of the tangent to x² + y² = a² at (x₁, y₁)?

xx₁ + yy₁ = a².

What is the condition for y = mx + c to be a tangent to x² + y² = a²?

c² = a²(1 + m²).

What is the length of the tangent from a point to a circle?

It is √S₁, where S₁ is the value of the circle's equation at that point.

What is the condition for two circles to be orthogonal?

2g₁g₂ + 2f₁f₂ = c₁ + c₂.

How many common tangents do two externally touching circles have?

Three.