Graphing Circles

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Question 1 of 4

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 4

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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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 $($ $($ $- 1$ $-1$ $,$ $,$ $-5$ $)$ .

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

Complete the sketch
Question 3 of 4

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

$2x^2+4x+2y^2-16y+18=0$
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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

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

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

Rewrite the given equation in standard form.

Start by dividing both sides of the equation by 2.

$2x^2+4x+2y^2-16y+18=0$

$x^2+2x$ $+$ $y^2-8y$ $+9=0$

Now complete the squares.

Start by adding constants to the expressions $x^2+2x$ and $y^2-8y$ .

The constant to add to $x^2+2x$ is half the coefficient of $x$ squared.

$(\frac{1}{2}\times2)^2=1^2=$ $1$

The constant to add to $y^2-8y$ is half the coefficient of $y$ squared.

$(\frac{1}{2}\times(-8))^2=4^2=$ $16$

Since you are adding $1$ and $16$ to the left-hand side, add $1$ $+$ $16$ $=17$ to the right-hand side

$x^2+2x+1$ $+$ $y^2-8x+16$ $+9=17$

Now use the facts that $x^2+2x+1=(x+1)^2$ and $y^2-8x+16=(y-4)^2$ to rewrite these parts in the equation.

$(x+1)^2$ $+$ $(y-4)^2$ $+9=17$

Subtract $9$ from both sides of the equation and finish rewriting in standard form.

$(x+1)^2+(y-4)^2=8$

$(x-$ $(-1)$ $)^2+(y-$ $4$ $)^2=$ $\sqrt{8}^2$

In this equation $h=-1$ $,$ $k=4$ and $r=\sqrt{8}$

The centre is $($ $-1$ $,$ $4$ $)$ and the radius is $\sqrt{8}\approx 2.828$

To sketch the circle first plot the centre $($ $-1$ $,$ $4$ $)$ .

Since the radius is $2.828$ , plot the points which are $2.828$ units above, below, left and right of the centre.

Complete the sketch
Question 4 of 4

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

$x^2+y^2-4x+6y-3=0$
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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

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

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

Rewrite the left-hand side so the terms with the same variables are together.

$x^2-4x$ $+$ $y^2+6y$ $-3=0$

Now complete the squares.

Start by adding constants to the expressions $x^2-4x$ and $y^2+6y$ .

The constant to add to $x^2-4x$ is half the coefficient of $x$ squared.

$(\frac{1}{2}\times (-4))^2=2^2=$ $4$

The constant to add to $y^2+6y$ is half the coefficient of $y$ squared.

$(\frac{1}{2}\times 6)^2=3^2=$ $9$

Since you are adding $4$ and $9$ to the left-hand side, add $4$ $+$ $9$ $=13$ to the right-hand side

$x^2-4x+4$ $+$ $y^2+6y+9$ $-3=13$

Now use the facts that $x^2-4x+4=(x-2)^2$ and $y^2+6x+9=(y+3)^2$ to rewrite these parts in the equation.

$(x-2)^2$ $+$ $(y+3)^2$ $-3=13$

Add $3$ to both sides of the equation and finish rewriting in standard form.

$(x-2)^2+(y+3)^2=16$

$(x-$ $2$ $)^2+(y-$ $(-3)$ $)^2=$ $4^2$

In this equation $h=2$ $,$ $k=-3$ and $r=4$

The centre is $($ $2$ $,$ $-3$ $)$ and the radius is $4$ .

To sketch the circle first plot the centre $($ $2$ $,$ $-3$ $)$ .

Since the radius is $4$ , plot the points which are $4$ units above, below, left and right of the centre.

Complete the sketch

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