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Middle Years

11.04 Circles (Extended)

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

A circle has its centre at the origin and a radius of $9$9 units.

  1. Plot the graph for the given circle.

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  2. Write the equation of the circle.

Question 2

Consider the circle $x^2+y^2=25$x2+y2=25.

  1. State the coordinates of the centre of the circle.

  2. What is the radius of the circle?

  3. Find the $x$x values of the $x$x-intercepts. Write all solutions on the same line separated by a comma.

  4. Find the $y$y values of the $y$y-intercepts. Write all solutions on the same line separated by a comma.

  5. Graph the circle.

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Question 3

The circle $x^2+y^2=4^2$x2+y2=42 is translated $4$4 units down. Which of the following diagrams shows the new location of the circle?

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    A

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    B

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    A

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    B

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