Question

circles in the coordinate plane

which of the following is the graph of
\\((x - 1)^2 + (y + 2)^2 = 4\\)?

Explanation:

Identify the center and radius from the equation

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

$$ (x - 1)^2 + (y + 2)^2 = 4 \implies h = 1,\ k = -2,\ r = 2 $$

$$ \text{Center: } (1, -2),\ \text{Radius: } r = \sqrt{4} = 2 $$

Determine the key boundary points of the circle

$$ \text{Top: } (1, -2 + 2) = (1, 0) $$

$$ \text{Bottom: } (1, -2 - 2) = (1, -4) $$

$$ \text{Right: } (1 + 2, -2) = (3, -2) $$

$$ \text{Left: } (1 - 2, -2) = (-1, -2) $$

Match with the correct graph option

• Option 1 (left): Center is at \((1, -2)\) and radius is \(2\).
• Option 2 (middle): Center is at \((-1, 2)\) and radius is \(2\).
• Option 3 (right): Center is at \((1, 2)\) and radius is \(2\).

Answer:

• (A) The first graph (left) with center at \((1, -2)\) and radius \(2\) (Correct answer)
• (B) The second graph (middle) with center at \((-1, 2)\) and radius \(2\)
• (C) The third graph (right) with center at \((1, 2)\) and radius \(2\)