Question

complete the square and write the equation of the circle in standard form. then find the center and radius of the circle and graph the equation. x² + y² + 5x - 2y - 1 = 0

Explanation:

Step1: Group x - terms and y - terms

$(x^{2}+5x)+(y^{2}-2y)=1$

Step2: Complete the square for x - terms

For the expression $x^{2}+5x$, we add $(\frac{5}{2})^{2}=\frac{25}{4}$ to both sides of the equation.
$(x^{2}+5x+\frac{25}{4})+(y^{2}-2y)=1 + \frac{25}{4}$

Step3: Complete the square for y - terms

For the expression $y^{2}-2y$, we add $(-1)^{2} = 1$ to both sides of the equation.
$(x^{2}+5x+\frac{25}{4})+(y^{2}-2y + 1)=1+\frac{25}{4}+1$

Step4: Rewrite in standard form

$(x+\frac{5}{2})^{2}+(y - 1)^{2}=\frac{4 + 25+4}{4}=\frac{33}{4}$
The standard - form of the circle equation is $(x+\frac{5}{2})^{2}+(y - 1)^{2}=\frac{33}{4}$.
The center of the circle is $(-\frac{5}{2},1)$ and the radius $r=\frac{\sqrt{33}}{2}$.

Answer:

Standard form: $(x+\frac{5}{2})^{2}+(y - 1)^{2}=\frac{33}{4}$; Center: $(-\frac{5}{2},1)$; Radius: $\frac{\sqrt{33}}{2}$