Question

Given the general form of the equation of a circle, x2 + y2 + ax + by + c = 0, explain how to (i) write the equation in standard form, and (ii) how to graph the circle in standard form. Also demonstrate (i) and (ii) by providing an example.Can you explain how to graph the equation of a circle. Also demonstrate it by providing an example that have numbers for a, b and c in x2 + y2 + ax + by + c = 0.

Answer 1

It's easiest to do these in reverse. Let us first start with standard form, expand, and then compare with the original. Here is the equation in standard form for a circle with radius \(r\) located at \((x_0,y_0)\).\[\left(x-x_0\right)^2 + \left(y-y_0\right)^2 = r^2\]This hopefully should be second-nature to you. Let us expand this.\[x^2-2x_0x+x_0^2+y^2-2y_0y+y_0^2=r^2\]Rearranging gives us the following. I have underbraced the parts analogous to the form that you have been given in the problem. \[x^2+y^2\underbrace{-2x_0}_{a}x\underbrace{-2y_0}_{b}y+\underbrace{x_0^2+y_0^2-r^2}_c=0\]Below, I have written exclusively the parts representing a, b, and c. \[a=-2x_0\]\[b=-2y_0\]\[c=x_0^2+y_0^2-r^2\]We have three equations and three unknowns, so we can solve for x_0, y_0, and r. Note: The expression for c was found by substituting the found values of x_0 and y_0.\[x_0 = -\frac{a}{2}\]\[y_0 = -\frac{b}{2}\]\[r^2 = \frac{a^2}{4}+\frac{b^2}{4}-c\]Plugging these values in gives us the answer to part (i).\[\boxed{\left(x+\frac{a}{2}\right)^2+\left(y+\frac{b}{2}\right)^2=\frac{a^2}{4}+\frac{b^2}{4}-c}\]This describes a circle centered at \(\left(-\frac{a}{2},-\frac{b}{2}\right)\) and with radius \(\sqrt{\frac{a^2}{4}+\frac{b^2}{4}-c}\). I trust you can reason out the rest. If you need more help or clarification, just ask.

Answer 2

wow! Ok- well I got all the way through accept the part explaining how to graph the equation of a circle. Also demonstrate it by providing an example that have numbers for a, b and c in x2 + y2 + ax + by + c = 0.

Answer 3

Well give me three value for a, b, and c. Then find the center and how big the radius is. That's how you know how to graph it.

Answer 4

Center is (-a/2, -b/2)

Radius is √a2 + b2 -4c/4 if a2 + b2 -4c ≥ 0

Answer 5

Right, now just give me three numerical values of a, b, and c that work and give me numerical values for the center and radius.

Answer 6

I have no idea. lol

Answer 7

thats my problem....the last part of the question.

Answer 8

Give me three numerical values for a, b, and c of your choosing.

Answer 9

2, 1,0

Answer 10

thus where is the center? by the formulas we have?

Answer 11

ok

Answer 12

is the center at 0?

Answer 13

No. Center is (-a/2, -b/2), and you told me a=2 and b=1.

Answer 14

oh well see, im confused. lol

Answer 15

if a=2, then what is -a/2?

Answer 16

a= -2 ?

Answer 17

no... plug in and evaluate.

Answer 18

-2/2 = 0 ?! wth

Answer 19

wait = -1 ?

Answer 20

yes

Answer 21

how about -b/2?

Answer 22

b= -1

Answer 23

no...

Answer 24

b=1, thus, -b/2=?

Answer 25

-1/2= -0.5

Answer 26

yep, so our center is (-1,-.5). what's the radius?

Answer 27

radius is R ??

Answer 28

yes...what's the numerical value?

Answer 29

not sure how to get that

Answer 30

plug in and evaluate the numbers just as before

Answer 31

i cant do it, im frustrated...just dont know

Answer 32

\[r=\sqrt{\frac{a^2}{4}+\frac{b^2}{4}-c}\]Every time you see an a, put in your value of a. Every time you see a b, put in your value of b. Every time you see a c, put in your value of c. You'll be left with an expression of all numbers on the right hand side. Simplify it until you get a single number. This is your radius r.

Answer 33

but what is c?

Answer 34

you said it was 0

Answer 35

oh- so i can pick any numbers and it will be right? lol

Answer 36

as long as they're the same ones you used for the centers...use the same ones you picked before

Answer 37

That's what it means to given an example.

Answer 38

*give

Answer 39

ok i came up with 1.25

Answer 40

You forgot to square root it. It doesn't come out nice so we can just say \[r=\sqrt{1.25}\]

Answer 41

ok.

Answer 42

A calculator will give you r=1.11803399.

Answer 43

ok so how do i put my answer down then for this last question...

Answer 44

you give your example for a, b, and c, as well as your found center and radius.

Answer 45

can you show me how to write it though

Answer 46

HELLO!