Question

identifying the radius given an equation in general form
what is the radius of a circle given by the equation $x^2 + y^2 - 2x + 8y - 47= 0?$
radius = $square$ units

Explanation:

Step1: Group x and y terms

$x^2 - 2x + y^2 + 8y = 47$

Step2: Complete the square for x

For $x^2-2x$: $(-\frac{2}{2})^2=1$, so add 1 to both sides:
$x^2 - 2x + 1 + y^2 + 8y = 47 + 1$

Step3: Complete the square for y

For $y^2+8y$: $(\frac{8}{2})^2=16$, so add 16 to both sides:
$(x^2 - 2x + 1) + (y^2 + 8y + 16) = 48 + 16$

Step4: Rewrite in standard form

$(x-1)^2 + (y+4)^2 = 64$

Step5: Identify radius

Standard circle form: $(x-h)^2+(y-k)^2=r^2$, so $r^2=64$, solve for $r$:
$r = \sqrt{64}$

Answer:

8 units