Question

1. (0 points) find the center and radius of the circle with equation:

\\x^2 + y^2 - 8x + 10y + 5 = 0\\
and graph the circle.

\\((h, k) = \\)

\\(r = \\)

Explanation:

Step 1: Group terms

Group the \(x\) and \(y\) terms and move the constant.

$$(x^2 - 8x) + (y^2 + 10y) = -5$$

Step 2: Complete the square

Add the square of half the coefficients to both sides.

$$(x^2 - 8x + 16) + (y^2 + 10y + 25) = -5 + 16 + 25$$

Step 3: Write in standard form

Simplify both sides into standard circle equation form.

$$(x - 4)^2 + (y + 5)^2 = 36$$

Step 4: Identify center and radius

Compare with \((x - h)^2 + (y - k)^2 = r^2\).

$$(h, k) = (4, -5), \quad r = \sqrt{36} = 6$$

Answer:

Center \((h, k) = (4, -5)\)
Radius \(r = 6\)