Please teach me how to answer this SAT math question. ---The equation $x^2 + (y - 2)^2 = 36$ represents circle A. Circle B is obtained by shifting circle A down 4 units in the xy-plane. Which of the following equations represents circle B? A $x^2 + (y + 2)^2 = 36$ B $x^2 + (y - 6)^2 = 36$ C $(x - 4)^2 + (y - 2)^2 = 36$ D $(x + 4)^2 + (y - 2)^2 = 36$

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Let's solve this SAT math question about circles. We're given that the equation x squared plus y minus 2 squared equals 36 represents circle A. Circle B is obtained by shifting circle A down 4 units in the xy-plane. We need to find which of the given options represents the equation of circle B. First, let's understand what the equation of circle A tells us. To solve this problem, we need to understand the standard form of a circle equation. The standard form is (x minus h) squared plus (y minus k) squared equals r squared, where (h, k) is the center of the circle and r is its radius. Looking at circle A's equation, x squared plus (y minus 2) squared equals 36, we can identify that the center is at (0, 2) and the radius is 6, which is the square root of 36. This is because when we compare to the standard form, h equals 0 and k equals 2. Now, let's apply the transformation. We're told that circle B is obtained by shifting circle A down 4 units in the xy-plane. When we shift a circle down, only the y-coordinate of its center changes. The center of circle A is at (0, 2). Shifting down 4 units means we subtract 4 from the y-coordinate. So the center of circle B will be at (0, 2 minus 4), which equals (0, negative 2). The radius remains unchanged at 6. Using the standard form of a circle equation with the new center, we get (x minus 0) squared plus (y minus negative 2) squared equals 36. This simplifies to x squared plus (y plus 2) squared equals 36. Now let's compare our derived equation for circle B with the given options. We found that the equation of circle B is x squared plus (y plus 2) squared equals 36. Looking at the options, we can see that option A matches exactly with our derived equation. Option B has (y minus 6) squared, which would place the center at (0, 6), not at (0, negative 2). Option C has (x minus 4) squared, which would shift the circle to the right, not down. Option D has (x plus 4) squared, which would shift the circle to the left. Therefore, the correct answer is option A: x squared plus (y plus 2) squared equals 36. Let's summarize what we've learned. The standard form of a circle equation is (x minus h) squared plus (y minus k) squared equals r squared, where (h,k) is the center and r is the radius. When a circle is shifted down by d units, only the y-coordinate of its center changes: k becomes k minus d. For circle A with equation x squared plus (y minus 2) squared equals 36, the center is at (0,2) and the radius is 6. Shifting it down 4 units gives circle B with center at (0, negative 2), resulting in the equation x squared plus (y plus 2) squared equals 36. When solving problems involving geometric transformations, focus on how the transformation affects the key parameters in the equation. This approach will help you solve similar problems efficiently.