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.

$2x^2+4x+2y^2-16y+18=0$
- 1.
- 2.
- 3.
- 4.
Hint

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Correct

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Incorrect

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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.

$x^2+y^2-4x+6y-3=0$
- 1.
- 2.
- 3.
- 4.
Hint

Help Video

Correct

Well Done!

Incorrect

Help Video

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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