Which of the following is the equation of the circle with centre (-1.5, 0.5) and radius 3?
A. 2x^2 + 6x + 2y^2 - 2y - 13 = 0
B. 2x^2 - 6x + 2y^2 + 2y - 13 = 0
C. 2x^2 - 6x + 2y^2 - 2y - 13 = 0
D. 2x^2 + 6x + 2y^2 - 2y - 23 = 0
E. @x^2 + 6x + 2y^2 - 2y -14 = 0
Well, let's see here. The equation of a circle with center (h, k) and radius r is given by (x - h)^2 + (y - k)^2 = r^2.
So, in this case, with a center of (-1.5, 0.5) and a radius of 3, the equation should be (x + 1.5)^2 + (y - 0.5)^2 = 9.
Hmm... none of the options listed seem to match exactly. Looks like we've got some mathematical mischief going on here!
But if I had to choose the closest one, option A, 2x^2 + 6x + 2y^2 - 2y - 13 = 0, seems to come the closest. So, I guess we'll go with that one.
Remember, sometimes math can be a circus!
To determine the equation of a circle, we can use the standard form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle and r is the radius.
Given that the center of the circle is (-1.5, 0.5) and the radius is 3, we can substitute these values into the standard form:
(x - (-1.5))^2 + (y - 0.5)^2 = 3^2
Simplifying:
(x + 1.5)^2 + (y - 0.5)^2 = 9
Expanding:
(x + 1.5)(x + 1.5) + (y - 0.5)(y - 0.5) = 9
(x^2 + 3x + 2.25) + (y^2 - y + 0.25) = 9
Rearranging terms:
x^2 + 3x + 2.25 + y^2 - y + 0.25 = 9
x^2 + 3x + y^2 - y + 2.5 = 9
Combining like terms:
x^2 + 3x + y^2 - y - 6.5 = 0
Comparing this equation with the given options, we can see that the correct equation is:
C. 2x^2 - 6x + 2y^2 - 2y - 13 = 0
To determine the equation of a circle in standard form, we use the formula (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius. In this case, the center is (-1.5, 0.5) and the radius is 3.
Plugging in these values, we get:
(x - (-1.5))^2 + (y - 0.5)^2 = 3^2
(x + 1.5)^2 + (y - 0.5)^2 = 9
Expanding and rearranging, we have:
x^2 + 3x + 2.25 + y^2 - y + 0.25 = 9
x^2 + 3x + y^2 - y + 2.5 = 9
x^2 + 3x + y^2 - y - 6.5 = 0
Comparing this equation to the given options, we see that the correct answer is:
C. 2x^2 - 6x + 2y^2 - 2y - 13 = 0
C(-1.5,0.5), P(x,y)
r^2 = (x+1.5)^2 + (y-0.5)^2 = 3^2
x^2+3x+2.25 + y^2-y+0.25 = 9
x^2+3x + y^2-y = 9-2.25-0.25 = 6.5
x^2+3x=(3/2)^2 + y^2-y+(-1/2)^2 = 6.5
x^2+3x+9/4 + y^2-y+1/4 = 26/4+9/4+1/4
x^2+3x+9/4 + Y^2-y+1/4 = 36/4
Multiply both sides by 4:
4x^2+12x+9 + 4y^2-4y+1 = 36
4x^2+12x + 4y^2-4y-26 = 0
Divide both sides by 2:
2x^2+6x + 2y^2-2y-13 = 0