Tangent of a Circle

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A tangent is a line that aligns with something at one point. Therefore, a tangent of a circle is a line that aligns with the circle at one point.

For example, $x = 3$ is a tangent to the circle $x^{2} + y^{2} = 9$

because it touches the circle once at the point (3, 0).

Tangents of a circle Graphical depiction example StudySmarter Tangent to a circle

A tangent differs from a secant because a secant is a line that intercepts the circles in two places.

How to find the equation of the tangent of a circle

Many questions are concerned with finding an equation for the tangent of a circle. To find the equation of the tangent of a circle, you need to understand how the tangent relates to the radius of the circle. The tangent relates to the radius between an exterior point (a point on the circle) and the circle centre. The exterior point acts as the point of intersection between the radius of the circle and the tangent.

Finding the gradient of the radius between the circle centre and the exterior point

The first step for finding the equation of a tangent of a circle at a specific point is to find the gradient of the radius of the circle. You need the radius between the circle centre and the exterior point because it will be perpendicular to the tangent. This is because this radius of the circle is acting as a normal line to the tangent.

To find the gradient of the radius of the circle, you substitute the points in the circle centre and the exterior point into the gradient formula:

$G r a d i e n t = \frac{C h a n g e i n y}{C h a n g e i n x}$

A circle with the equation $x^{2} + y^{2} = 9$ touches a tangent at (3, 0). What is the gradient of the radius of the circle which is perpendicular to the given tangent?

As there are no constants attached to the x or y variables, you can interpret that the circle centre is (0,0)
Graphically, this information can be represented like this

Tangent of a circle Tangent of a circle worked example StudySmarter

You are finding the equation of the line (radius) connecting the circle centre and exterior point on the circle (between the blue and green points).

Inputting your coordinates of the circle centre and the exterior point into the gradient formula:

G r a d i e n t o f t h e c i r c l e r a d i u s = \frac{c h a n g e i n y}{c h a n g e i n x} = \frac{0 - 0}{3 - 0} = 0 .

Therefore, the gradient of the circle radius is 0.

Finding the gradient of the tangent of the circle

One of our Circle Theorems is the Equation of a Perpendicular Bisector. This is where the tangent of a circle perpendicularly intercepts with the radius of the circle. Therefore, to find the gradient of the tangent, you need to do the negative reciprocal of the gradient of the radius of the circle. If the gradient of the radius is m, then the gradient of the tangent is $\frac{- 1}{m}$ .

What is the gradient of the tangent line to a circle when the gradient of the radius of the same circle has a gradient of $\frac{1}{4}$ ?

The negative reciprocal of the tangent (¼) is $- \frac{1}{\frac{1}{4}} = - 4$

Therefore, the gradient of the tangent of the circle is -4.

Finding the equation of the tangent of the circle

Once we have the exterior point and the gradient of the tangent, we can use an equation formula to find the equation of the tangent.

There are three formulas to help you form Equations for the tangent to the circle:

$y = m x + c$

$y - y_{1} = m (x - x_{1})$

$A x + B y = C$

The first two are much easier to use than the third. So if you are asked to put your final answer into the third form, use the first or second formula, then rearrange into that form.

A tangent touches circle A at (5, 6). What is the equation of a tangent of circle A, when the gradient of the radius is $- \frac{1}{5}$ ?

To find the gradient of the tangent, you do the negative reciprocal so $- \frac{1}{- \frac{1}{5}} = 5$
Using $y = m x + c$ , you could substitute in (5, 6) and the gradient (5) and then rearrange

$6 = 5 (5) + c$

$6 - 25 = c = - 19$

$y = 5 x - 19$

Or, using $y - y_{1} = m (x - x_{1})$ and (5, 6)

y - 6 = 5 (x - 5)

These are the same Equations:

y - 6 = 5 (x - 5)

$y - 6 = 5 x - 25$

$y = 5 x - 19$

Therefore, the equation of the tangent to circle A can be written as

y = 5 x - 19

and

y - 6 = 5 (x - 5)

Worked example to create an equation for the tangent of a circle

Question: Circle 1 has the equation $x^{2} + y^{2} = 25$ . A tangent aligns itself to circle 1 at point (4, -3). What is the equation for this circle?

Step 1: Find the gradient of the radius of the circle

As there are no constants attached to x and the y variables, you can conclude the circle centre is (0,0).
To find the gradient of the radius of the circle, you substitute your coordinates into the gradient formula.

$G r a d i e n t o f t h e r a d i u s o f t h e c i r c l e = \frac{c h a n g e i n y}{C h a n g e i n x} = \frac{- 3 - 0}{4 - 0} = \frac{- 3}{4}$

Therefore, the gradient of the circle at point (4, -3) is $\frac{- 3}{4}$ .

Step 2: Find the gradient of the tangent of the circle

To find the gradient of the tangent of the circle, you do the inverse reciprocal of the gradient of the radius of the circle.

$G r a d i e n t o f t h e \tan g e n t o f t h e c i r c l e = \frac{- 1}{m} = - \frac{1}{\frac{- 3}{4}} = \frac{4}{3}$

Therefore, the gradient of the tangent of the circle is $\frac{4}{3}$

Step 3: Find the equation for the tangent of the circle

I will be using the $y - y_{1} = m (x - x_{1})$ formula, substituting the gradient of the tangent of the circle ( $\frac{4}{3}$ ) and the exterior point (4, -3).

$E q u a t i o n o f t h e \tan g e n t o f t h e c i r c l e = y + 3 = \frac{4}{3} (x - 4)$

Graphically, we can see this as:

Tangent of a circle Graphical depiction example StudySmarter

Tangent of a Circle - Key takeaways

Tangents of a circle are lines that are touching the circumference at a given point and can be shown through a linear equation.
You need the circle equation and the point of intersection to find the equation of the tangent.
To find the tangent of a circle, you need to work out the gradient of the radius from the centre to the point of interception. The gradient of the radius of the circle will act as the normal line perpendicular to the tangent. Then, you do the negative reciprocal of the gradient of the radius to get the gradient of the tangent. Finally, you substitute the point of intersection into a linear equation formula to get the equation of the tangent.

Frequently Asked Questions about Tangent of a Circle

The tangent of a circle is a line that touches the circumference of the circle at one point. A tangent differs from a secant, because a secant aligns with two points of the circle. A tangent is expressed as a linear equation.

First you find the gradient of the radius using the centre of the circle and the point of interception. The gradient of the radius acts like a normal line perpendicular to the tangent. Then, to find the gradient of the tangent, you do the negative reciprocal of the gradient of the radius. Finally, substitute the point of intersection and the tangent gradient into one of the linear equation formulas to get the equation of the tangent to the circle.

Flashcards in Tangent of a Circle14 Start learning

What is a tangent?

A tangent is a line which touches the circle at a specific point

What are the steps to finding the tangent of a circle at a specific point?

Find the gradient of the radius using the centre of the circle and the point of interception; find the gradient of the tangent using the radius; using a linear equation formula, the gradient of the tangent and the point of interception, create a linear equation for the tangent

What does the radius of a circle act as when we are finding the tangent?

A normal line perpendicular to the tangent

How do you find the gradient of the tangent using the gradient of the radius of the circle?

Find the negative reciprocal of the gradient of radius of the circle

Circle B has the equation (x+4)²+(y+2)²=10. A tangent touches Circle B at point (-1, -1). What is the equation of the tangent?

y=3x+2

Circle C has the equation (x-3)²+(y-2)²=20. A tangent touches Circle C at point (7, 4). What is the equation of the tangent?

y=-2x+18

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Tangent of a Circle

How to find the equation of the tangent of a circle

Finding the gradient of the radius between the circle centre and the exterior point

Finding the gradient of the tangent of the circle

Finding the equation of the tangent of the circle

Worked example to create an equation for the tangent of a circle

Tangent of a Circle - Key takeaways

Frequently Asked Questions about Tangent of a Circle

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Tangent of a Circle

How to find the equation of the tangent of a circle

Finding the gradient of the radius between the circle centre and the exterior point

Finding the gradient of the tangent of the circle

Finding the equation of the tangent of the circle

Worked example to create an equation for the tangent of a circle

Tangent of a Circle - Key takeaways

Frequently Asked Questions about Tangent of a Circle

What is the tangent of a circle?

How do you find the equation of the tangent of a circle?

Learn with 14 Tangent of a Circle flashcards in the free StudySmarter app

Join over 22 million students in learning with our StudySmarter App

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