Question

Find an equation for the circle that has center (−3, 4) and passes through the point (4, −2).

Answer 1

To find the equation of the circle with center (-3, 4) and passing through (4, -2), we determine the radius using the distance formula and plug it into the standard form of a circle's equation, resulting in (x + 3)^2 + (y - 4)^2 = 85.

Step-by-step explanation:

To find the equation of a circle, we can use the standard form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. Given the center at (-3, 4) and a point on the circle (4, -2), we first calculate the radius by finding the distance between these two points using the distance formula:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

r = \sqrt{(4 - (-3))^2 + (-2 - 4)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}

So the equation of the circle is:

(x + 3)^2 + (y - 4)^2 = (\sqrt{85})^2

(x + 3)^2 + (y - 4)^2 = 85

This is the equation of the circle with center (-3, 4) that passes through the point (4, -2).

Answer 2

First use the coordinate of the center to start building your equation. The equation of a circle recall is

`{(x - h)}^(2) + {(y - k)}^(2) = {r}^(2)`
With the center point being (h,k).

Once you put in the centers coordinates, your equation should be,

`{(x + 3)}^(2) + {(y - 4)}^(2) = {r}^(2)`
However we still need to find the radius. We can do this using the second point given, by putting it into the equation and solving for the radius.

`{( 4 + 3)}^(2) + {( - 2 - 4)}^(2) = {r}^(2)`
Solving this gives us the final part of the equation.

`85 = {r}^(2)`
And we can finally put our full equation together,

`{(x + 3)}^(2) + {(y - 4)}^(2) = 85`