Question

the equation of a circle is given below. identify the center and the radius. then graph the circle.
$2x^2 + 2y^2 - 8x - 42 = 0$
center: ( , )
radius:

Explanation:

Step1: Divide by 2 to simplify

Divide every term by 2 to reduce coefficients:
$$x^2 + y^2 - 4x - 21 = 0$$

Step2: Rearrange and group variables

Group x-terms and move constant to right:
$$(x^2 - 4x) + y^2 = 21$$

Step3: Complete the square for x

Add $(\frac{-4}{2})^2=4$ to both sides:
$$(x^2 - 4x + 4) + y^2 = 21 + 4$$
$$(x-2)^2 + y^2 = 25$$

Step4: Match to circle standard form

Standard form: $(x-h)^2+(y-k)^2=r^2$, where $(h,k)$ is center, $r$ is radius.
Compare to get $h=2$, $k=0$, $r^2=25$ so $r=5$.

Answer:

Center: $(2, 0)$
Radius: $5$

To graph the circle:

1. Plot the center point at $(2, 0)$ on the coordinate grid.
2. From the center, count 5 units up, down, left, and right to plot the points $(2,5)$, $(2,-5)$, $(-3,0)$, $(7,0)$.
3. Draw a smooth, closed curve connecting these four points to form the circle.