Can someone help me understand how to do this:
I need to find two solutions to the problem
(x-2)^2 + (y+3)^2 = 9
i tried simplifying it down to
x^2 - 4x + y^2 + 6y + 4 = 0
but I don't know if I'm on the right track. Can anyone help?
To me that doesn't seem like a complete question. You have 1 equation with 2 unknowns. You need a minimum of 2 equations to find 2 unknowns.
Another way of looking at it is that the above equation is the equation for a circle with center (2,-3) and radius 3.
So asking for two solutions for a circle doesn't make much sense.
Are you sure that's the whole question?
Hope that helps.
Of course! I can help you understand how to find two solutions to the problem.
The equation you have given is a circle equation in standard form: (x - h)^2 + (y - k)^2 = r^2. In this form, the center of the circle is at the point (h, k), and the radius of the circle is r.
For the equation (x-2)^2 + (y+3)^2 = 9, we can see that the center of the circle is at the point (2, -3), and the radius is √9 = 3.
To find two solutions to this problem, we need to identify two points that lie on the circle.
Solution 1:
Let's start by considering a point that is 3 units to the right of the center. Since the x-coordinate of the center is 2, the x-coordinate of the point we're looking for would be 2 + 3 = 5. To find the y-coordinate, we substitute this value of x into the equation and solve for y:
(5 - 2)^2 + (y + 3)^2 = 9
3^2 + (y + 3)^2 = 9
9 + (y + 3)^2 = 9
(y + 3)^2 = 0
Solving for y, we get:
y + 3 = 0
y = -3
So, the first solution is (5, -3).
Solution 2:
Now, let's consider a point that is 3 units down from the center. Since the y-coordinate of the center is -3, the y-coordinate of the point we're looking for would be -3 - 3 = -6. To find the x-coordinate, we substitute this value of y into the equation and solve for x:
(x - 2)^2 + (-6 + 3)^2 = 9
(x - 2)^2 + (-3)^2 = 9
(x - 2)^2 + 9 = 9
(x - 2)^2 = 0
Solving for x, we get:
x - 2 = 0
x = 2
Therefore, the second solution is (2, -6).
So, the two solutions to the problem (x-2)^2 + (y+3)^2 = 9 are (5, -3) and (2, -6).