Circles and Ellipses

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Circles with translations

Lesson

We've looked at circles that are centred around the origin. The general form

$x^2+y^2=r^2$`x`2+`y`2=`r`2

But what happens when the centre of the circle is not at the origin?

Circles are everywhere. From clocks to wheels to rings to coins to DVDs, you won't go through a day without seeing a circle. Given the great number of objects that resemble circles, it would be helpful to be able to graph them to study their properties. The following interactive allows you to explore the standard form equation of a circle. It shows how the equation changes as the coordinates of the centre ($h$`h`, $k$`k`) and the radius $r$`r` change. To move the circle, drag the sliders for $h$`h` and $k$`k`, the centre coordinates of the circle, while to change the radius, just drag the $r$`r` slider. You'll notice that, regardless of the values, the equation will always be in the form $\left(x-h\right)^2+\left(y-k\right)^2=r^2$(`x`−`h`)2+(`y`−`k`)2=`r`2.

Equation of a Circle in Standard Form

$\left(x-h\right)^2+\left(y-k\right)^2=r^2$(`x`−`h`)2+(`y`−`k`)2=`r`2

where $\left(h,k\right)$(`h`,`k`) is the coordinates of the centre of the circle

and $r$`r` is the radius of the circle

Equation of a Circle in General Form

The standard form of a circle can also be rearranged and written in general form. The general form of a circle is:

$x^2+y^2+ax+by+c=0$`x`2+`y`2+`a``x`+`b``y`+`c`=0

A circle has its centre at $\left(3,3\right)$(3,3) and a radius of $6$6 units.

a) Plot the graph for the given circle.

b) Write the equation of the circle.

The equation of a circle is given by $\left(x+4\right)^2+\left(y-2\right)^2=25$(`x`+4)2+(`y`−2)2=25.

a) Find the coordinates of the centre of this circle.

b) What is the radius of the circle?

c) Plot the graph for the given circle.

Write down the equation of the new circle after $x^2+y^2=49$`x`2+`y`2=49 is translated:

a) $5$5 units upwards

b) $5$5 units downwards

c) $5$5 units to the right

d) $5$5 units to the left and $6$6 units upwards

Apply the geometry of conic sections

Apply the geometry of conic sections in solving problems