Victorian Curriculum Year 10A - 2020 Edition 5.09 Circles
Lesson

Graphs of equations of the form $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(xh)2+(yk)2=r2 (where $h$h, $k$k, and $r$r are any number and $r\ne0$r0) are called circles. The circle defined by $x^2+y^2=1$x2+y2=1. It has a centre at $\left(0,0\right)$(0,0) and a radius of $1$1 unit

### Transformations of circles

A circle can be vertically translated by increasing or decreasing the $y$y-values by a constant number. However, the $y$y-value together with the translation must be squared together. So to translate $x^2+y^2=1$x2+y2=1 up by $k$k units gives us $x^2+\left(y-k\right)^2=1$x2+(yk)2=1. Vertically translating up by $2$2 ($x^2+\left(y-2\right)^2=1$x2+(y2)2=1) and down by $2$2 ($x^2+\left(y+2\right)^2=1$x2+(y+2)2=1)

Similarly, a circle can be horizontally translated by increasing or decreasing the $x$x-values by a constant number. However, the $x$x-value together with the translation must be squared together. That is, to translate $x^2+y^2=1$x2+y2=1 to the left by $h$h units we get $\left(x+h\right)^2+y^2=1$(x+h)2+y2=1. Horizontally translating left by $2$2 ($\left(x+2\right)^2+y^2=1$(x+2)2+y2=1) and right by $2$2 ($\left(x-2\right)^2+y^2=1$(x2)2+y2=1)

Notice that the centre of the circle $x^2+y^2=1$x2+y2=1 is at $\left(0,0\right)$(0,0). Translating the circle will also translate the centre by the same amount. So the centre of $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(xh)2+(yk)2=r2 is at $\left(h,k\right)$(h,k).

A circle can be scaled both vertically and horizontally by changing the value of $r$r. In fact, $r$r is the radius of the circle Expanding by a scale factor of $2$2 ($x^2+y^2=4$x2+y2=4) and compressing by a scale factor of $2$2 ($x^2+y^2=\frac{1}{4}$x2+y2=14)

Summary

The graph of an equation of the form $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(xh)2+(yk)2=r2 is a circle.

Circles have a centre at $\left(h,k\right)$(h,k) and a radius of $r$r

Circles can be transformed in the following ways (starting with the circle defined by $x^2+y^2=1$x2+y2=1):

• Vertically translated by $k$k units: $x^2+\left(y-k\right)^2=1$x2+(yk)2=1
• Horizontally translated by $h$h units: $\left(x-h\right)^2+y^2=1$(xh)2+y2=1
• Scaled (both vertically and horizontally) by a scale factor of $r$r: $x^2+y^2=r^2$x2+y2=r2

#### Practice questions

##### Question 1

Consider the circle with equation $\left(x-0.4\right)^2+\left(y+3.8\right)^2=2$(x0.4)2+(y+3.8)2=2.

1. What is the centre of the circle?

Give your answer in the form $\left(a,b\right)$(a,b).

2. What is the radius of the circle?

##### Question 2

A circle has its centre at $\left(3,-2\right)$(3,2) and a radius of $4$4 units.

1. Plot the graph for the given circle.

2. Write the equation of the circle in general form.

##### Question 3

Consider the equation of a circle given by $x^2+4x+y^2+6y-3=0$x2+4x+y2+6y3=0.

1. Rewrite the equation of the circle in standard form.

2. What are the coordinates of the centre of this circle?

Give your answer in the form $\left(a,b\right)$(a,b).

3. What is the radius of this circle?

### Outcomes

#### VCMNA339

Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital technology as appropriate

#### VCMNA359 (10a)

Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations.