Question

Match the circle equations in general form with their corresponding equations in standard form.

Tiles

x2 + y2− 4x + 12y − 20 = 0

(x − 6)2+ (y − 4)2 = 56

x2 + y2+ 6x − 8y − 10 = 0 

(x − 2)2+ (y + 6)2 = 60

3x2 + 3y2 + 12x + 18y − 15 = 0

(x + 2)2+ (y + 3)2 = 18

5x2 + 5y2 − 10x + 20y − 30 = 0

(x + 1)2+ (y − 6)2 = 46

2x2 + 2y2 − 24x − 16y − 8 = 0

x2 + y2+ 2x − 12y − 9 =

Answer 1

case A)
`x^(2) +y^(2) -4x+12y-20=0`

Convert to standard form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`(x^(2)-4x)+(y^(2)+12y)=20`

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

`(x^(2)-4x+4)+(y^(2)+12y+36)=20+4+36`

`(x^(2)-4x+4)+(y^(2)+12y+36)=60`

Rewrite as perfect squares

`(x-2)^(2)+(y+6)^(2)=60`

therefore

the answer case A) is

`x^(2) +y^(2) -4x+12y-20=0` ----->
`(x-2)^(2)+(y+6)^(2)=60`

case B)
`x^(2) +y^(2) +6x-8y-10=0`

Convert to standard form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`(x^(2)+6x)+(y^(2)-8y)=10`

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

`(x^(2)+6x+9)+(y^(2)-8y+16)=10+9+16`

`(x^(2)+6x+9)+(y^(2)-8y+16)=35`

Rewrite as perfect squares

`(x+3)^(2)+(y-4)^(2)=35`

therefore

the answer case B) is

`x^(2) +y^(2) +6x-8y-10=0` ----->
`(x+3)^(2)+(y-4)^(2)=35`

case C)
`3x^(2) +3y^(2) +12x+18y-15=0`

Convert to standard form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`(3x^(2)+12x)+(3y^(2)+18y)=15`

Factor the leading coefficient of each expression

`3(x^(2)+4x)+3(y^(2)+6y)=15`

`(x^(2)+4x)+(y^(2)+6y)=5`

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

`(x^(2)+4x+4)+(y^(2)+6y+9)=5+4+9`

`(x^(2)+4x+4)+(y^(2)+6y+9)=18`

Rewrite as perfect squares

`(x+2)^(2)+(y+3)^(2)=18`

therefore

the answer case C) is

`3x^(2) +3y^(2) +12x+18y-15=0` ----->
`(x+2)^(2)+(y+3)^(2)=18`

case D)
`5x^(2) +5y^(2) -10x+20y-30=0`

Convert to standard form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`(5x^(2)-10x)+(5y^(2)+20y)=30`

Factor the leading coefficient of each expression

`5(x^(2)-2x)+5(y^(2)+4y)=30`

`(x^(2)-2x)+(y^(2)+4y)=6`

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

`(x^(2)-2x+1)+(y^(2)+4y+4)=6+1+4`

`(x^(2)-2x+1)+(y^(2)+4y+4)=11`

Rewrite as perfect squares

`(x-1)^(2)+(y+2)^(2)=11`

therefore

the answer case D) is

`3x^(2) +3y^(2) +12x+18y-15=0` ----->
`(x-1)^(2)+(y+2)^(2)=11`

case E)
`2x^(2) +2y^(2) -24x-16y-8=0`

Convert to standard form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`(2x^(2)-24x)+(2y^(2)-16y)=8`

Factor the leading coefficient of each expression

`2(x^(2)-12x)+2(y^(2)-8y)=8`

`(x^(2)-12x)+(y^(2)-8y)=4`

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

`(x^(2)-12x+36)+(y^(2)-8y+16)=4+36+16`

`(x^(2)-12x+36)+(y^(2)-8y+16)=56`

Rewrite as perfect squares

`(x-6)^(2)+(y-4)^(2)=56`

therefore

the answer case E) is

`2x^(2) +2y^(2) -24x-16y-8=0` ----->
`(x-6)^(2)+(y-4)^(2)=56`

case F)
`x^(2) +y^(2)+2x-12y-9=0`

Convert to standard form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`(x^(2)+2x)+(y^(2)-12y)=9`

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

`(x^(2)+2x+1)+(y^(2)-12y+36)=9+1+36`

`(x^(2)+2x+1)+(y^(2)-12y+36)=46`

Rewrite as perfect squares

`(x+1)^(2)+(y-6)^(2)=46`

therefore

the answer case F) is

`x^(2) +y^(2)+2x-12y-9=0` ----->
`(x+1)^(2)+(y-6)^(2)=46`