a circle is drawn on a coordinate plane with the center point at (-1, 3) and a radius of 5. derive the equation of the circle in standard form, using the given information and the pythagorean theorem.

Question

a. (x + 1)^2 + (y - 3)^2 = 25
b. (x - 3)^2 + (y - 1)^2 = 25
c. (x - 1)^2 + (y - 3)^2 = 25
d. (x + 3)^2 + (y - 1)^2 = 25

Answer 1

To derive the equation of a circle in standard form, we can use the formula:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

Given:

• Center \((-1, 3)\): Here, \(h = -1\) and \(k = 3\).
• Radius \(r = 5\).

Now, we can plug these values into the formula:

1. First, substitute \(h\) and \(k\): \[ (x - (-1))^2 + (y - 3)^2 = 5^2 \] This simplifies to: \[ (x + 1)^2 + (y - 3)^2 = 25 \]

Thus, the equation of the circle in standard form is:

\[ (x + 1)^2 + (y - 3)^2 = 25 \]

Therefore, the correct answer is:

a. (x + 1)^2 + (y - 3)^2 = 25