Graph Circles in Standard Form

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Question 1 of 2

1. Question
Identify the centre and radius, then sketch the following circle.

${(x + 2)}^{2} + {(y - 4)}^{2} = 25$ $(x+2)^2+(y-4)^2=25$
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In order to graph the circle, first find the centre and radius

The centre and radius can be read from the standard form equation

$(x -$ $(x-$ $h$ $h$ $)^{2} + (y -$ $)^2+(y-$ $k$ $k$ $)^{2} =$ $)^2=$ $r^{2}$ $r^2$

where the centre is $($ $($ $h$ $h$ $,$ $,$ $k$ $k$ $)$ $)$ and the radius is $r$ $r$

Rewrite the given equation in standard form

$(x -$ $(x-$ $(- 2)$ $(-2)$ $)^{2} + (y -$ $)^2+(y-$ $4$ $4$ $)^{2} =$ $)^2=$ $5^{2}$ $5^2$

In this equation $h = - 2$ $h=-2$ $,$ $,$ $k = 4$ $k=4$ and $r = 5$ $r=5$

The centre is $($ $($ $- 2$ $-2$ $,$ $,$ $4$ $4$ $)$ $)$ and the radius is $5$ $5$

To sketch the circle first plot the centre $($ $($ $- 2$ $-2$ $,$ $,$ $4$ $4$ $)$ $)$ .

Since the radius is $5$ $5$ , plot the points which are $5$ $5$ units above, below, left and right of the centre.

Complete the sketch
Question 2 of 2

2. Question
Find the radius of the circle which has centre $(- 1, - 5)$ $(-1,-5)$ and passes through $(2, - 1)$ $(2,-1)$ . Then sketch this circle.
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Hint

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Correct

Excellent!

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In order to graph the circle, first find the radius

The centre and radius can be read from the standard form equation

$(x -$ $(x-$ $h$ $h$ $)^{2} + (y -$ $)^2+(y-$ $k$ $k$ $)^{2} =$ $)^2=$ $r^{2}$ $r^2$

where the centre is $($ $($ $h$ $h$ $,$ $,$ $k$ $k$ $)$ $)$ and the radius is $r$ $r$ .

The centre of our circle is $($ $($ $- 1$ $-1$ $,$ $,$ $- 5$ $-5$ $)$ $)$ , so in the standard equation $h = - 1$ $h=-1$ $and$ $and$ $k = - 5$ $k=-5$

$(x -$ $(x-$ $(- 1)$ $(-1)$ $)^{2} + (y -$ $)^2+(y-$ $- 5$ $-5$ $)^{2} =$ $)^2=$ $r^{2}$ $r^2$

${(x + 1)}^{2} + {(y + 5)}^{2} =$ $(x+1)^2+(y+5)^2=$ $r^{2}$ $r^2$

We know the point $($ $($ $2$ $2$ $,$ $,$ $- 1$ $-1$ $)$ $)$ , lies on the circle, so we can substitute $x = 2$ $x=2$ and $y = - 1$ $y=-1$ into the equation and solve to find $r$ $r$ .

$($ $($ $2$ $2$ $+ 1)^{2} + ($ $+1)^2+($ $- 1$ $-1$ $+ 5)^{2} =$ $+5)^2=$ $r^{2}$ $r^2$

${(3)}^{2} + {(4)}^{2} =$ $(3)^2+(4)^2=$ $r^{2}$ $r^2$

$9 + 16 =$ $9+16=$ $r^{2}$ $r^2$

$25 =$ $25=$ $r^{2}$ $r^2$

$r = 5$ $r=5$

To sketch the circle first plot the centre $($ $($ $- 1$ $-1$ $,$ $,$ $- 5$ $-5$ $)$ $)$ .

Since the radius is $5$ $5$ , plot the points which are $5$ $5$ units above, below, left and right of the centre.

Complete the sketch

Graph Circles in Standard Form

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