Question

Find the equation for the circle with center (-4,-5) and passing through (4,-2)

Answer 1

(x + 4)² + (y + 5)² = 73

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

here (h, k) = (- 4, - 5), thus

(x - (- 4))² + (y - (- 5))² = r², that is

(x + 4)² + (y + 5)² = r²

The radius is the distance from the centre to a point on the circle.

Calculate r using the distance formula

r = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (- 4, - 5) and (x₂, y₂ ) = (4, - 2)

r =
`√((4+4)^2+(-2+5)^2)`

=
`√(8^2+3^2)`

=
`√(64+9)` =
`√(73)` ⇒ r² = 73, thus

(x + 4)² + (y + 5)² = 73 ← equation of circle

Answer 2

Answer:
`(x+4)^(2) +(y+5)^(2) = 73`

Explanation:

the formula for finding equation of circle is given as :

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

where ( a,b) is the coordinate of the center and r is the radius

The radius is the distance between the point and the center , the formula for calculating the distance between two points is given by :

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

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

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

`d = \sqrt{8^(2)+3^(2)}`

`d = √(73)`

since "r" is the distance , then

`r = √(73)`

The the equation of the circle , using the formula
`(x-a)^(2)+(y-b)^(2) = r^(2)` , will be

`(x+4)^(2) +(y+5)^(2) = 73`