Victorian Curriculum Year 10A - 2020 Edition
topic badge
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?

    Give an exact answer.

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.

    Loading Graph...

  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.

What is Mathspace

About Mathspace