Circle Graphs and Tangents Worksheets | Questions and Revision | MME

# Circle Graphs and Tangents Worksheets, Questions and Revision

Level 8 Level 9

## What you need to know

Circle graphsare another type of graph you need to know about. Questions involving circle graphs are some of the hardest on the course.

You need to be able to plot them as well as calculate the equation of tangents.

Make sure you are happy with the following topics

## What are Circle Graphs

Circle graphs for a circle, they have the general equation

$\textcolor{blue}{r}^2 = \textcolor{red}{x}^2 + \textcolor{limegreen}{y}^2$

e.g:

$\textcolor{blue}{4}^2 = \textcolor{red}{x}^2 + \textcolor{limegreen}{y}^2$

which is the same as

$\textcolor{blue}{16} = \textcolor{red}{x}^2 + \textcolor{limegreen}{y}^2$

Remember: $r$ is the radius of the circle formed around centre $(0,0)$

## Example

Find the equation of the tangent to the circle defined by

$x^2 + y^2 = \textcolor{red}{25}$

at the point $(3, 4)$, shown in on the axes below.

[4 marks]

Firstly, we can recognise that because $\textcolor{red}{5^2} = \textcolor{red}{25}$, the radius of this circle is $\textcolor{red}{5}$

Now, the first step of this process is to find the gradient of the radius which goes from the centre of the circle to the point $(3,4)$.

This can be seen in the diagram below.

The tangent is perpendicular to the radius at that point (one of our circle theorems), and if you know the gradient of a line you can obtain the gradient of the perpendicular by taking the negative reciprocal of it.

So, the gradient of the line that goes from the origin to $(3,4)$ is

$\text{Gradient } = \dfrac{\text{change in } y}{\text{change in }x} = \dfrac{4 - 0}{3 - 0} = \dfrac{4}{3}$.

Taking the negative reciprocal of this, we get

$\text{Gradient of tangent } = \textcolor{blue}{-\dfrac{3}{4}}$

Now we know the gradient, our straight-line equation must be $y = \textcolor{blue}{-\frac{3}{4}}x + c$, where $c$ is the y-intercept that we are yet to determine.

We know is that this tangent passes through the point $(3,4)$, so we can substitute these values of $x$ and $y$ into our straight-line equation and rearrange to find $c$. We get:

$4 = \textcolor{blue}{-\dfrac{3}{4}} \times 3 + c$

$4 = -\dfrac{9}{4} + c$

$\textcolor{limegreen}{c} = 4 + \dfrac{9}{4} = \textcolor{limegreen}{\dfrac{25}{4}}$

Now we’ve found $c$, we can express our equation of our tangent fully:

$y = \textcolor{blue}{-\dfrac{3}{4}}x + \textcolor{limegreen}{\dfrac{25}{4}}$

### Example Questions

We can see that the radius of this circle extends a distance of 10 away from the centre at $(0,0)$. Therefore, because $10^2 = 100$, the equation of the circle is

$x^2 + y^2 = 100$

First, we need to find the gradient of the line from the centre to $(12, 5)$.

$\text{Gradient of radius } = \dfrac{\text{change in } y}{\text{change in }x} = \dfrac{5 - 0}{12 - 0} = \dfrac{5}{12}$

Now, by observing that this line we’ve found the gradient of is a radius, and tangents are perpendicular to the radius, we can find the gradient of the tangent by taking the negative reciprocal of the answer we got above.

$\text{Gradient of tangent } = -\dfrac{12}{5}$

So, we know the straight-line equation for our tangent must be of the form

$y = -\dfrac{12}{5}x + c$,

where $c$ is the y-intercept which we must determine. To do this, we can substitute the values of $x = 12$ and $y = 5$ into the straight-line equation, since we know the line must pass through those coordinates. We get the following.

$5 = -\dfrac{12}{5} \times 12 + c$

$5 = -\dfrac{144}{5} + c$

$c = 5 + \dfrac{144}{5} = \dfrac{169}{5}$

$y = -\dfrac{12}{5}x + \dfrac{169}{5}$

First, we need to find the gradient of the line from the centre to $(-8, -7)$.

$\text{Gradient of radius } = \dfrac{\text{change in } y}{\text{change in }x} = \dfrac{-7 - 0}{-8 - 0} = \dfrac{7}{8}$

Now, by observing that this line we’ve found the gradient of is a radius, and tangents are perpendicular to the radius, we can find the gradient of the tangent by taking the negative reciprocal of the answer we got above.

$\text{Gradient of tangent } = -\dfrac{8}{7}$

So, we know the straight-line equation for our tangent must be of the form

$y = -\dfrac{8}{7}x + c$,

where $c$ is the y-intercept which we must determine. To do this, we can substitute the values of $x = -8$ and $y = -7$ into the straight-line equation, since we know the line must pass through those coordinates. We get the following.

$-7 =-\dfrac{8}{7} \times -8 + c$

$-7 = \dfrac{64}{7} + c$

$c=-\dfrac{113}{7}$

$y = -\dfrac{8}{7}-\dfrac{113}{7}$

Level 8-9

Level 8-9

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