Conic Sections Worksheets – Circles

Work with the equation of a circle: find its center and radius, complete the square and write the equation. Grades 11–12.

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Circles in the coordinate plane

Conic sections are the curves made by slicing a cone: circles, parabolas, ellipses and hyperbolas. These worksheets currently cover the circle, the conic students usually study first.

A circle with center (h,k)(h, k) and radius rr is the set of points at distance rr from the center. Written with the distance formula, that gives the standard form:

(x−h)2+(y−k)2=r2(x - h)^{2} + (y - k)^{2} = r^{2}
  • Easy — read the center and radius from an equation in standard form.
  • Medium — rewrite x2+y2+Ax+By=Cx^{2} + y^{2} + Ax + By = C in standard form by completing the square.
  • Hard — write the equation from the center and one point on the circle.

Worked examples

Easy: (x−2)2+(y+5)2=49(x - 2)^{2} + (y + 5)^{2} = 49

Match the pattern: x−2x - 2 gives h=2h = 2, and y+5=y−(−5)y + 5 = y - (-5) gives k=−5k = -5. The right side is r2=49r^{2} = 49, so r=7r = 7. Answer: center (2,−5)(2, -5), radius 7.

Medium: x2+y2−6x+4y=12x^{2} + y^{2} - 6x + 4y = 12

Group the xx-terms and the yy-terms, then complete each square. Half of −6-6 is −3-3, and (−3)2=9(-3)^{2} = 9; half of 4 is 2, and 22=42^{2} = 4. Add 9 and 4 to both sides:

(x2−6x+9)+(y2+4y+4)=12+9+4(x^{2} - 6x + 9) + (y^{2} + 4y + 4) = 12 + 9 + 4 (x−3)2+(y+2)2=25(x - 3)^{2} + (y + 2)^{2} = 25

Answer: center (3,−2)(3, -2), radius 5.

Hard: center (−1,4)(-1, 4), passing through (−1,7)(-1, 7)

The radius is the distance from the center to the point. Both points have x=−1x = -1, so the distance is 7−4=37 - 4 = 3.

(x+1)2+(y−4)2=9(x + 1)^{2} + (y - 4)^{2} = 9

Check your answers

Substitute the given point: (−1+1)2+(7−4)2=0+9=9(-1 + 1)^{2} + (7 - 4)^{2} = 0 + 9 = 9. The point lies on the circle.

Common mistakes

  • Taking the sign from the equation. (x−2)2(x - 2)^{2} means h=2h = 2, and (y+5)2(y + 5)^{2} means k=−5k = -5. The center's coordinates have the opposite signs.
  • Confusing rr and r2r^{2}. The right side is r2r^{2}. For =49= 49 the radius is 7, not 49.
  • Completing the square on one side only. The 9 and 4 added on the left must also be added on the right.
  • Writing rr instead of r2r^{2}. In the hard example the equation ends in =9= 9, not =3= 3.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

  1. (x − 2)2 + (y + 1)2 = 64
    Find center and radius
  2. (x − 6)2 + (y − 6)2 = 1
    Find center and radius
  3. (x − 4)2 + (y − 5)2 = 1
    Find center and radius

Medium

  1. x2 + y2 + 8x = 20
    Write in standard form; state center and radius
  2. x2 + y2 − 6x = −5
    Write in standard form; state center and radius
  3. x2 + y2 + 4x + 2y = 31
    Write in standard form; state center and radius

Hard

  1. Circle: center (2, -1), passes through (2, 1)
    Write the equation
  2. Circle: center (5, 3), passes through (10, 3)
    Write the equation
  3. Circle: center (4, 2), passes through (-2, 2)
    Write the equation
Show answers
  1. Center: (2, -1), r = 8
  2. Center: (6, 6), r = 1
  3. Center: (4, 5), r = 1
  4. (x + 4)² + y² = 36; Center: (-4, 0), r = 6
  5. (x − 3)² + y² = 4; Center: (3, 0), r = 2
  6. (x + 2)² + (y + 1)² = 36; Center: (-2, -1), r = 6
  7. (x − 2)² + (y + 1)² = 4
  8. (x − 5)² + (y − 3)² = 25
  9. (x − 4)² + (y − 2)² = 36

Tips for teachers and parents

  • Plot one circle from each worksheet on grid paper; seeing the center and radius makes the sign rule easy to remember.
  • Medium problems build on perfect-square trinomials from Factoring Polynomials. A quick review of (x−3)2=x2−6x+9(x - 3)^{2} = x^{2} - 6x + 9 helps.
  • The unit circle x2+y2=1x^{2} + y^{2} = 1 links this topic to trigonometry.

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