Question

A circle is centered at the point (5, -4) and passes through the point (-3, 2).

The equation of this circle is (x +_ )2 + (y +_ )2 = _

Answer 1

The first space is 3,

The second space is -2

The third space is 100

Explanation:

A circle with center (a, b) and has radius r has equation:
`(x-a)^(2) + (y-b)^(2) = r^(2)`

Now, if the circle passes through (-3, 2) and it has center on (5, -4). That means the radius of the circle will be the distance between points (-3, 2) and (5, -4).

`d = \sqrt{(x_(2) - x_(1) )^(2) + (y_(2) - y_(1) )^(2) }`

where d = formula of distance between points
`(x_(2), y_(2) ) \ and \ (x_(1), y_(1) )`

`d = \sqrt{(-3-5)^(2) + (-4-2)^(2) }\\ d = \sqrt{-8^(2) + -6^(2) }\\ d = √(64+36)\\ d = √(100)\\ d = radius \ of \ circle = 10`

Now, with r = 10, a = -3, b = 2

The equation of circle becomes:

`(x-(-3))^(2) + (y-2)^(2) = 10^(2) \\(x+3)^(2) + (y-2)^(2) = 100 \\`

The first space is 3,

The second space is -2

The third space is 100

Answer 2

The answer to your question is (x - 5)² + (x + 4) = 100

Explanation:

Data

Center = (5, -4)

Point = (-3, 2)

Process

1.- Calculate the length of the radius

Formula

d =
`\sqrt{(x2-x1)^(2) + (y2 - y1)^(2)}`

Substitution

d =
`\sqrt{(-3-5)^(2)+ (2 + 4)^(2)}`

Simplification

d =
`\sqrt{(-8)^(2)+ (6)^(2)}`

d=
`√(64 + 36)`

d =
`√(100)`

d = 10

2.- Get the equation of the line

h = 5 k = -4 r = 10

( x - 5)² + (y + 4)² = 10²

Simplification and result

(x - 5)² + (x + 4) = 100