Graph Circles in Expanded Form

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Question 1 of 2

1. 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 2 of 2

2. 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

Graph Circles in Expanded Form

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