Question

the equation of a circle is given below. identify the center and the radius. then graph the circle. $4x^{2}-36x + 4y^{2}+24y=-53$

Explanation:

Step1: Divide by 4 to simplify

$x^2 - 9x + y^2 + 6y = -\frac{53}{4}$

Step2: Complete square for x

$x^2 -9x = (x-\frac{9}{2})^2 - (\frac{9}{2})^2 = (x-\frac{9}{2})^2 - \frac{81}{4}$

Step3: Complete square for y

$y^2 +6y = (y+3)^2 - 3^2 = (y+3)^2 -9$

Step4: Substitute back and simplify

$(x-\frac{9}{2})^2 - \frac{81}{4} + (y+3)^2 -9 = -\frac{53}{4}$
$(x-\frac{9}{2})^2 + (y+3)^2 = -\frac{53}{4} + \frac{81}{4} +9$
$(x-\frac{9}{2})^2 + (y+3)^2 = \frac{28}{4} +9 =7+9=16$

Step5: Identify center and radius

Standard form: $(x-h)^2+(y-k)^2=r^2$, so $h=\frac{9}{2}$, $k=-3$, $r=4$

Answer:

Center: $(\frac{9}{2}, -3)$
Radius: $4$
(Graph: Plot center at $(\frac{9}{2}, -3)$ and draw a circle with radius 4 on the grid)