Question

Write the equation of the circle with center (-3,2) and (6,4) a point on the circle

Answer 1

Explanation:

Prerequisites:

You need to know:

The distance formula

The standard equation of a circle.

Distance formula:

d =
`√((X_2-X_1)^2+(Y_2 - Y_1)^2)`

The standard equation of a circle

`(x - h)^2 + (y - k)^2 = r^2`

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First, use the standard equation of the circle and plugin the information we know.

We know the center is (h, k) = (-3, 2)

h = -3

k = 2

`(x - h)^2 + (y - k)^2 = r^2`

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

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

Since we do not know our r value, we need to find r. We can find this value by using the distance formula. Remember, the radius (r) is the distance from the center of the circle to a point on the circle. Points we will use are (-3,2) (6,4)

`d = √((X_2-X_1)^2 + (Y_2 - Y_1)^2)`

`d = √((-3 - 6)^2 + (2 - 4)^2)`

`d = √((-9)^2 + (-2)^2)`

`d = √(81 + 4)`

`d = √(85) = 9.21954457`

Now we plugin 9.21954457 for r.

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

`(x + 3)^2 + (y - 2)^2 = 9.21954457^2`

`(x + 3)^2 + (y - 2)^2 = 84.9999999`

Answer:

`(x + 3)^2 + (y - 2)^2 = 84.99999999`

OR

`(x + 3)^2 + (y - 2)^2 = 85`

Answer 2

(x + 3)² + (y - 2)² = 85

the equation of a circle in standard form is

(x - a)² + (y - b)² = r²

where (a, b ) are the coordinates of the centre and r is the radius

the radius is the distance from the centre to the point (6, 4 ) on the circle

r = √(x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (- 3, 2 ) and (x₂, y₂ ) = (6, 4 )

r = √(6 + 3 )² + (4 - 2 )² = √(81 + 4 ) = √85 ⇒ r² = 85

(x + 3)² + (y - 2)² = 85 ← equation of circle