Question

question 7 (multiple choice worth 1 points) (07.03 lc) a circle is represented by the equation below: ((x + 2)^2 + (y - 4)^2 = 225) which statement is true? the circle is centered at ((-2, 4)) and has a radius of 15. the circle is centered at ((2, -4)) and has a diameter of 15. the circle is centered at ((2, -4)) and has a radius of 15. the circle is centered at ((-2, 4)) and has a diameter of 15.

Explanation:

The standard form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 1: Identify the center of the circle

For the given equation \((x + 2)^2 + (y - 4)^2 = 225\), we can rewrite \(x + 2\) as \(x - (-2)\). So, comparing with the standard form \((x - h)^2 + (y - k)^2 = r^2\), we have \(h = -2\) and \(k = 4\). Thus, the center of the circle is \((-2, 4)\).

Step 2: Identify the radius of the circle

We know that \(r^2 = 225\). To find \(r\), we take the square root of both sides: \(r=\sqrt{225} = 15\). The diameter \(d\) of a circle is related to the radius \(r\) by the formula \(d = 2r\). So, \(d = 2\times15 = 30\), not 15.

Now let's analyze each option:

• Option 1: The circle is centered at \((-2, 4)\) (correct center) and has a radius of 15 (correct radius, since \(r = 15\)).
• Option 2: The center is given as \((2, -4)\), which is incorrect. The correct center is \((-2, 4)\).
• Option 3: The center is \((2, -4)\) (incorrect) and the diameter is 15 (incorrect, diameter should be 30).
• Option 4: The center is \((-2, 4)\) (correct) but the diameter is 15 (incorrect, diameter is 30).

Answer:

A. The circle is centered at \((-2, 4)\) and has a radius of 15.