Tangents to a Circle (Analytical) | IEB Grade 12 Maths

Tangent questions join analytical geometry with the Euclidean result that a tangent is perpendicular to the radius. Expect to find tangent equations, tangent lengths, and the value of a constant that makes a line a tangent.

Equation of a tangent at a point on the circle

The tangent at $P$ is perpendicular to the radius $C P$ (tan $⊥$ radius).

Find the gradient of the radius from the centre $C$ to $P$ .
The tangent gradient is the negative reciprocal.
Use $y - y_{1} = m (x - x_{1})$ with the point $P$ .

Example. Tangent to $x^{2} + y^{2} = 25$ at $P (3; - 4)$ .

$m_{O P} = - \frac{4}{3} \Rightarrow m_{tan} = \frac{3}{4}$

$y + 4 = \frac{3}{4} (x - 3) \Rightarrow y = \frac{3}{4} x - \frac{25}{4}$

When is a line a tangent? Substitute the line into the circle. If the resulting quadratic has $Δ = 0$ , the line touches the circle at exactly one point.

Length of a tangent from an external point

If $T$ is outside the circle with centre $C$ and radius $r$ , and $T A$ is a tangent touching at $A$ , then $△ C A T$ has a right angle at $A$ :

$T A^{2} = T C^{2} - r^{2}$

Example. $T (7; 5)$ and the circle $(x - 1)^{2} + (y + 3)^{2} = 20$ .

$T C^{2} = (7 - 1)^{2} + (5 + 3)^{2} = 100$

$T A = 100 - 20 = 80 = 45$

Both tangents from $T$ have this length (tans from common pt).

Key terms

Point of contact: The point where the tangent touches the circle.
Negative reciprocal: For gradient $m$ , the perpendicular gradient $- \frac{1}{m}$ .
Discriminant: $Δ = b^{2} - 4 a c$ ; when it is 0, a line meets the circle once.

Common mistakes

Using the gradient of the radius as the gradient of the tangent.
Using the centre as the point in the tangent equation.
Forgetting to square the radius when finding tangent length.
Not checking that the given point actually lies on the circle.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

4 marks total

routine

Determine the equation of the tangent to $x^{2} + y^{2} = 25$ at $P (3; - 4)$ .

Worked solution

Hidden - try the question first

Solution

$m_{O P} = - \frac{4}{3}$
$m_{tan} = \frac{3}{4}$ (tan $⊥$ radius)
$y + 4 = \frac{3}{4} (x - 3)$
$y = \frac{3}{4} x - \frac{25}{4}$

Question 2

6 marks total

routine

The circle $(x - 1)^{2} + (y + 2)^{2} = 13$ has centre $C$ .

(a) Show that $P (3; 1)$ lies on the circle. (1)

(b) Determine the equation of the tangent to the circle at $P$ . (5)

Worked solution

Hidden - try the question first

Solution

(a) $(3 - 1)^{2} + (1 + 2)^{2} = 4 + 9 = 13$
(b) $C (1; - 2)$
$m_{C P} = \frac{1 + 2}{3 - 1} = \frac{3}{2}$
$m_{tan} = - \frac{2}{3}$
$y - 1 = - \frac{2}{3} (x - 3)$
$y = - \frac{2}{3} x + 3$

Question 3

6 marks total

complex

$T (7; 5)$ lies outside the circle $(x - 1)^{2} + (y + 3)^{2} = 20$ . $T A$ is a tangent to the circle at $A$ .

Calculate the length of $T A$ in simplest surd form and to two decimal places.

Worked solution

Hidden - try the question first

Solution

Centre $(3 - 1)^{2} + (1 + 2)^{2} = 4 + 9 = 13$ and $r^{2} = 20$
$T C^{2} = (7 - 1)^{2} + (5 + 3)^{2} = 100$
$\hat{C A T} = 90°$ (tan $⊥$ radius)
$T A^{2} = 100 - 20 = 80$
$T A = 45$
$\approx 8, 94$ units

Question 4

9 marks total

problem-solving

The line $y = 2 x + c$ is a tangent to the circle $x^{2} + y^{2} = 20$ .

(a) Determine the possible values of $c$ . (5)

(b) Determine the point of contact for each value of $c$ . (4)

Worked solution

Hidden - try the question first

Solution

(a) $(3 - 1)^{2} + (1 + 2)^{2} = 4 + 9 = 13$
$5 x^{2} + 4 c x + c^{2} - 20 = 0$
Tangent, so $Δ = 0$ : $(4 c)^{2} - 4 (5) (c^{2} - 20) = 0$
$m_{tan} = - \frac{2}{3}$
$c = 10$ or $c = - 10$
(b) $c = 10$ : $5 x^{2} + 40 x + 80 = 0 \Rightarrow (x + 4)^{2} = 0$
$x = - 4$ , $y = 2$ : $(- 4; 2)$
$c = - 10$ : $(x - 4)^{2} = 0$
$x = 4$ , $y = - 2$ : $(4; - 2)$

FAQs

How do I find the equation of a tangent to a circle?

Find the gradient of the radius to the point of contact, take its negative reciprocal, and use the point of contact in $y - y_{1} = m (x - x_{1})$ .

How do I show that a line is a tangent to a circle?

Substitute the line into the circle equation. If the quadratic has exactly one solution ( $Δ = 0$ ), the line is a tangent.

Which Euclidean theorems appear in analytical geometry?

Tangent perpendicular to radius, the angle in a semicircle, and the perpendicular from the centre bisecting a chord appear most often.

Tangents to a circle

Equation of a tangent at a point on the circle

Length of a tangent from an external point

Key terms

Common mistakes

Practice questions

FAQs

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