Explained: "Complete the square to transform the general form..."

High School
Geometry
High school geometry
Circles

hexacordoK

Answered question

2021-05-27

Complete the square to transform the general form equation below to standard form in order to determine the center and radius. $4 x^{2} + 4 y^{2} - 16 x - 24 y + 51 = 0$

Answer & Explanation

l1koV

Skilled2021-05-28Added 100 answers

Isolate the constant and group the variables:

(4 x^{2} - 16 x) + (4 y^{2} - 24 y) = - 51

Factor out 44:

4 (x^{2} - 4 x) + 4 (y^{2} - 6 y) = - 51

Complete the square:

4 (x^{2} - 4 x + 4) + 4 (y^{2} - 6 y + 9) = - 51 + 4 (4) + 4 (9)

4 {(x - 2)}^{2} + 4 {(y - 3)}^{2} = 1

Divide both sides by 4:

{(x - 2)}^{2} + {(y - 3)}^{2} = \frac{1}{4}

The standard form of an equation of a circle with center (h,k) and radius rr is:

{(x - h)}^{2} + {(y - k)}^{2} = r^{2}

Since h=2, k=3, and

r^{2} = \frac{1}{4} \to r = \frac{1}{2} r

, we have:
center: (2,3),radius:

\frac{1}{2}

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