Question

Rewrite the equation of each circle in Standard Form. Then graph.

Answer 1

(x- 4)²+(y+3)² = 3²

Explanation:

We have given the equation:

x²+y²-8x +6y +16 = 0

We have to rewrite the equation of circle in standard form.

The standard form of equation of circle is :

(x-c)²+(y-d)² = r²

Where r is radius and (c,d) is center.

x²-8x+y² +6y = -16

x² - 2(x)(4)+(4)²+ y²+6y = -16 +(4)²

(x- 4)²+y²+2(y)(3)+(3²) = -16+16+3²

(x- 4)²+(y+3)² = 3²

It is a circle of radius 3 and center (4,-3)

Answer 2

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

Explanation:

The standard form for the equation of a circumference is:

`(x-a) ^ 2 + (y-b) ^ 2 = r ^ 2`

Where:

(a, b) is the center of the circumference

r is the radius

In this problem we have the equation of the following circumference, and we want to convert it to the standard form:

`x ^ 2 + y ^ 2 - 8x + 6y +16 = 0`

The first thing we must do to transform this equation into the standard form is to use the square completion technique.

The steps are shown below:

1. Group all the same variables:

`(y ^ 2+6y) + (x ^ 2 - 8x) = -16`

2. Take the coefficient that accompanies the variable x. In this case the coefficient is -8. Then, divide by 2 and the result elevate it to the square.

We have:

`(-8)/(2) = -4\\\\((-8)/(2)) ^ 2 = 16`

Then do the same for the coefficient that accompanies the variable y

`(6)/(2) = 3\\\\((6)/(2)) ^ 2 = 9`

3. Add on both sides of the equality the terms obtained in the previous step:

`(y ^ 2+6y +9) + (x ^ 2 - 8x +16) = -16 + 9 + 16\\\\(y ^ 2+6y +9) + (x ^ 2 - 8x +16) = 9`

4. Factor the resulting expression, and you will get:

`(x-4)^2 +(y+3)^2 = 9`

Write the equation in the standard form:

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

Then, the center is the point (4, -3) and the radius is r = 3.

Observe the attached image