Find the equation for the circle with center ​(3​,-2​) and passing through ​(​-3,4​).

Answer 1

Hi, Jenna!

The equation of the circle can be expressed in the following form: (x - a)² + (y - b)² = r², where (a,b) is the center of the circle and r is the radius.

In this example, we are given the coordinates of the center of the circle and a point on the circle itself. We can start by plugging our center, (3, -2) into our general formula and simplifying:

(x - 3)² + (y - (-2))² = r²

(x - 3)² + (y + 2)² = r²

Next, we need to find the value of r, or the radius. To do so, we can use the fact that our point on the circle, (-3, 4), must satisfy our equation. Therefore, we can plug these values into the x and y in our equation to solve for r:

((-3) - 3)² + (4 + 2)² = r²

(-6)² + 6² = r²

36 + 36 = r²

r² = 72

Now, of note, we could stop here if we wanted to. Our equation has a r² in it, and not an r, so we could plug this into our equation for the circle to get our final answer:

(x - 3)² + (y + 2)² = 72

This is the equation of a circle with center (3, -2) and passing through the point (-3, 4)

If you needed to calculate the radius itself, you could do one extra step and take the square root of both sides of r² = 72 to get r = 6√2 (the radius is always a positive value).

I hope this explanation was helpful, and please don't hesitate to reach out if you have any further questions!

Answer 2

In general the equation for a circle is (x-h)^2+(y-k)^2=r^2 where (h, k) is the center of the circle and r is the radius. In your problem you are given h and k, so the only real challenge here is to find the radius r.

There are three methods you could use.

1. distance formula,
2. Pythagorean theorem
3. Use the given point and center in the original equation to solve for r.

Good luck!

Answer 3

a circle's general equation is

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

where (h,k) = the center, r= radius

for (h,k) = (3,-2)

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

plug in (-3,4) to solve for r^2

(-3-3)^2 + (4+2)^2 = r^2

36+36 = 72 =r^2

(x-3)^2 +(y+2)^2 = 72 is the equation with the given center through the given point