Question

Question #a46b5

Answer 1

`(x - 67/26)^2+(y - (-17/26))^2= (5/26sqrt370)^2`

Explanation:

Use the standard Cartesian equation of a circle, `(x-h)^2+(y-k)^2=r^2`, and the points, `(0,2),(1,-4),(2,3)`, to write 3 equations:

`(0-h)^2+(2-k)^2=r^2" [1]"`
`(1-h)^2+(-4-k)^2=r^2" [2]"`
`(2-h)^2+(3-k)^2=r^2" [3]"`

Expand the squares:

`h^2+4-4k+k^2=r^2" [1.1]"`
`1-2h+h^2+16+8k+k^2=r^2" [2.1]"`
`4-4h+h^2+9-6k+k^2=r^2" [3.1]"`

Subtract equation [1.1] from equation [2.1]:

`1 - 2h +h^2-h^2+16-4+8k+4k+k^2-k^2 = r^2-r^2`

Combine like terms and mark as equation [4]:

`13-2h+12k = 0" [4]"`

Subtract equation [1.1] from equation [3.1]:

`4-4h+h^2-h^2+9-4-6k+4k+k^2-k^2=r^2-r^2`

Combine like terms and mark as equation [5]:

`9-4h-2k=0" [5]"`

Multiply equation [5] by 6 and add to equation [4]:

#13+54-2h-24h+12k-12k=0

Combine like terms:

`67-26h`

`h = 67/26`

Substitute the value for h into equation [5] and solve for k:

`9-4(67/26)-2k=0`

`2k = 9-4(67/26)`

`k = 9/2-2(67/26)`

`k = -17/26`

Substitute the values of h and k into equation [1] and solve for r:

`(67/26)^2+(2+17/26)^2=r^2`

`r^2 = (67^2+69^2)/26^2`

`r = sqrt9250/26 = 5/26sqrt370`

Substitute the values of h, k, and r into the standard form:

`(x - 67/26)^2+(y - (-17/26))^2= (5/26sqrt370)^2`