Question

x² + y² - 4x + 8y + 19 = 0 is the equation of a circle with center (h, k) and radius r for: h = and k = and r = graph the circle.

Explanation:

Step1: Complete the square for \(x\) terms

Group \(x\) - terms: \(x^{2}-4x=(x - 2)^{2}-4\) (using \((a - b)^2=a^{2}-2ab + b^{2}\), here \(a = x\), \(b = 2\)).

Step2: Complete the square for \(y\) terms

Group \(y\) - terms: \(y^{2}+8y=(y + 4)^{2}-16\) (using \((a + b)^2=a^{2}+2ab + b^{2}\), here \(a = y\), \(b = 4\)).

Step3: Rewrite the circle equation

Substitute into the original equation \(x^{2}+y^{2}-4x + 8y+19 = 0\):
\((x - 2)^{2}-4+(y + 4)^{2}-16+19 = 0\).
Simplify to \((x - 2)^{2}+(y + 4)^{2}=1\).

Answer:

\(h = 2\)
\(k=-4\)
\(r = 1\)