Coordinate Geometry and Conics · Lesson 3

Circle equations and completing the square

Standard form. The circle with center (h,k)(h, k) and radius rr is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. Read the center with the opposite signs: (x+4)2(x + 4)^2 means h=−4h = -4. The right side is r2r^2, so the radius is its square root.

General form. An equation such as x2+y2+8x−2y−8=0x^2 + y^2 + 8x - 2y - 8 = 0 hides the center. Move the constant to the right, then complete the square for x and for y: take half of each linear coefficient, square it, and add that number to both sides. Then read the center and radius from the standard form.

Inside, on or outside. Put the point into the left side of the standard form and compare the result with r2r^2: less means inside, equal means on the circle, greater means outside. This is the distance formula without the square root.

Worked example Easy

A circle with center (-3, 2)−8−7−6−5−4−3−2−1123−3−2−11234567xy(−3, 2)(1, 2)
A circle with center (−3,2)(-3, 2) A circle in the standard (x, y) coordinate plane. Its center is the labelled point (−3, 2). The labelled point (1, 2) lies on the circle, 4 units to the right of the center, so the radius is 4. The circle reaches from x = −7 to x = 1 and from y = −2 to y = 6.

The circle shown in the standard (x, y) coordinate plane has center (−3,2)(-3, 2) and passes through the point (1,2)(1, 2). Which equation represents the circle?

  1. (x−3)2+(y+2)2=16(x - 3)^2 + (y + 2)^2 = 16
  2. (x+3)2+(y−2)2=4(x + 3)^2 + (y - 2)^2 = 4
  3. (x+3)2+(y−2)2=8(x + 3)^2 + (y - 2)^2 = 8
  4. (x+3)2+(y−2)2=16(x + 3)^2 + (y - 2)^2 = 16Answer

How to solve it

  1. The radius is the distance from (−3,2)(-3, 2) to (1,2)(1, 2): 1−(−3)=41 - (-3) = 4.
  2. Use (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 with h=−3h = -3, k=2k = 2 and r=4r = 4.
  3. (x+3)2+(y−2)2=16(x + 3)^2 + (y - 2)^2 = 16.

Why each choice is right or wrong

  • A. Incorrect. (x−3)2+(y+2)2=16(x - 3)^2 + (y + 2)^2 = 16 flips the signs of the center, so it describes a circle centered at (3,−2)(3, -2).
  • B. Incorrect. (x+3)2+(y−2)2=4(x + 3)^2 + (y - 2)^2 = 4 puts the radius on the right side instead of its square. The right side is r2=16r^2 = 16.
  • C. Incorrect. (x+3)2+(y−2)2=8(x + 3)^2 + (y - 2)^2 = 8 doubles the radius instead of squaring it. The right side is 42=164^2 = 16, not 2⋅42 \cdot 4.
  • D. Correct. The center is (−3,2)(-3, 2) and the radius is 4, so the equation is (x+3)2+(y−2)2=42=16(x + 3)^2 + (y - 2)^2 = 4^2 = 16.

Worked example Medium

In the standard (x, y) coordinate plane, what are the center and radius of the circle with equation x2+y2+8x−2y−8=0x^2 + y^2 + 8x - 2y - 8 = 0?

  1. Center (−4,1)(-4, 1), radius 5Answer
  2. Center (−4,1)(-4, 1), radius 222\sqrt{2}
  3. Center (4,−1)(4, -1), radius 5
  4. Center (−8,2)(-8, 2), radius 5

How to solve it

  1. Move the constant: (x2+8x)+(y2−2y)=8(x^2 + 8x) + (y^2 - 2y) = 8.
  2. Half of 8 is 4 and 42=164^2 = 16; half of −2-2 is −1-1 and (−1)2=1(-1)^2 = 1. Add 16 and 1 to both sides.
  3. (x+4)2+(y−1)2=25(x + 4)^2 + (y - 1)^2 = 25, so the center is (−4,1)(-4, 1) and the radius is 5.

Why each choice is right or wrong

  • A. Correct. Completing the square gives (x+4)2+(y−1)2=8+16+1=25(x + 4)^2 + (y - 1)^2 = 8 + 16 + 1 = 25, so the center is (−4,1)(-4, 1) and the radius is 25=5\sqrt{25} = 5.
  • B. Incorrect. A radius of 22=82\sqrt{2} = \sqrt{8} forgets to add 16 and 1 to the right side as well, so it treats 8 as r2r^2.
  • C. Incorrect. Center (4,−1)(4, -1) copies the signs of +8x+8x and −2y-2y. The square (x+4)2(x + 4)^2 means h=−4h = -4, and (y−1)2(y - 1)^2 means k=1k = 1.
  • D. Incorrect. Center (−8,2)(-8, 2) takes the opposites of the coefficients 8 and −2-2 without halving them. Each coordinate of the center is half of that: (−4,1)(-4, 1).

Trap Signs of the center, and r squared

(x+4)2+(y−1)2=25(x + 4)^2 + (y - 1)^2 = 25 has center (−4,1)(-4, 1) and radius 5. The usual slips: flipping the center's signs, reporting 25 as the radius, or forgetting to add the completing-the-square numbers to the right side too.

Calculator When a calculator is faster

If you test online, type the equation exactly as given, such as x^2+y^2+8x-2y-8=0, into the built-in Desmos calculator and read the center and radius off the graph, then match them to a choice. For inside or outside, plot the point too. A handheld calculator that graphs only y = … equations needs the equation solved for y, so on paper completing the square is usually quicker.