Question

X^2 + y^2 +2x+6y=26

what is the center

what is the radius

please show calculations/steps

Answer 1

Complete the squares:

`(x {}^(2) + 2x + 1) - 1 = (x + 1) {}^(2) - 1`

`(y {}^(2) + 6y + 9) - 9 = (y + 3) {}^(2) - 9`

`(x + 1) {}^(2) - 1 + (y + 3) {}^(2) - 9 = 26`

`(x + 1) {}^(2) + (y + 3) {}^(2) = 26 + 10`

`(x + 1) {}^(2) + (y + 3) {}^(2) = 36`

General form:

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

Center coordinates:

I ( a , b )

I ( -1 , -3 )

Radius:

`r {}^(2) = 36`

`r = √(36) = 6`

Answer 2

centre = (- 1, - 3 ) , radius = 6

Explanation:

the equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k ) are the coordinates of the centre and r is the radius

given

x² + y² + 2x + 6y = 26 ( collect x and y terms )

x² + 2x + y² + 6y = 26

using the method of completing the square

add ( half the coefficient of the x/ y terms )² to both sides

x² + 2(1)x + 1 + y² + 2(3)y + 9 = 26 + 1 + 9

(x + 1)² + (y + 3)² = 36 ← in standard form

with centre = (- 1, - 3 ) and r =
`√(36)` = 6