# 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