LEMMA

Analytic geometry

Circles and conic sections

ALEKS placement

Completing the square turns a general second-degree equation into recognizable center form.

What this covers

(x−h)2+(y−k)2=r2
  • Group the x-terms and the y-terms, and move the constant across.
  • Complete the square on each group.
  • Add the same amounts to the right side.
  • Read the center (h, k) and the radius or axis lengths.

Worked example

Worked example

Find the center and radius of x² + y² - 6x + 4y - 12 = 0.

  1. (x2−6x)+(y2+4y)=12Group and move the constant.
  2. (x−3)2+(y+2)2=25Add 9 and 4 to both sides.
  3. center (3,−2),r=5Read them off the standard form.
Another worked example

For (x−1)²/9 + (y+2)²/4 = 1, find the center, semi-axis lengths, and foci.

  1. (h,k)=(1,−2),a=3,b=2The larger denominator is under x, so the major axis is horizontal. Take square roots of the denominators.
  2. c2=a2−b2=9−4=5For an ellipse, focal distance is shorter than the semi-major axis.
  3. foci=(1±√5,−2)Move c units in each horizontal direction from the center.

A common mistake

A mistake Lemma catches

From (x−3)2+(y+2)2=25

h=−3h=3

“The center is where both squares are 0. (x − 3)² is 0 at x = 3: (3 − 3)² = 0. Reading the sign as written, x = −3 gives (−3 − 3)² = 36, not 0.”

Try one

Sample problem

Find the radius of the circle.

(x−1)2+(y+2)2=4

Practice conic sections free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this