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Difficulty: MediumConic Sections

A circle in the standard (x,y)(x, y) coordinate plane has center (3,−4)(3, -4) and passes through the point (6,0)(6, 0). Which of the following is an equation of this circle?

  1. x2+y2−6x+8y=0x^2 + y^2 - 6x + 8y = 0Answer
  2. B
    x2+y2+6x−8y=0x^2 + y^2 + 6x - 8y = 0
  3. C
    x2+y2−6x+8y+20=0x^2 + y^2 - 6x + 8y + 20 = 0
  4. D
    x2+y2−18=0x^2 + y^2 - 18 = 0
  5. E
    x2+y2−6x+8y−24=0x^2 + y^2 - 6x + 8y - 24 = 0

Answer

x2+y2−6x+8y=0x^2 + y^2 - 6x + 8y = 0
The standard equation of a circle is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center and rr is the radius. Given the center is (3,−4)(3, -4), the equation becomes (x−3)2+(y+4)2=r2(x - 3)^2 + (y + 4)^2 = r^2. Since the circle passes through (6,0)(6, 0), we substitute these coordinates to find r2r^2: (6−3)2+(0+4)2=32+42=9+16=25(6 - 3)^2 + (0 + 4)^2 = 3^2 + 4^2 = 9 + 16 = 25. The equation is therefore (x−3)2+(y+4)2=25(x - 3)^2 + (y + 4)^2 = 25. Expanding this equation gives x2−6x+9+y2+8y+16=25x^2 - 6x + 9 + y^2 + 8y + 16 = 25. Combining constant terms yields x2+y2−6x+8y+25=25x^2 + y^2 - 6x + 8y + 25 = 25. Subtracting 25 from both sides results in the general form equation x2+y2−6x+8y=0x^2 + y^2 - 6x + 8y = 0.

Step-by-Step Solution

1
Calculate the radius squared of the circle using the distance formula between the center (3,−4)(3, -4) and the point on the circle (6,0)(6, 0).
r2=(6−3)2+(0−(−4))2=32+42=9+16=25r^2 = (6 - 3)^2 + (0 - (-4))^2 = 3^2 + 4^2 = 9 + 16 = 25
The distance between the center and any point on the circle is equal to the radius of the circle.
2
Write the standard form of the circle's equation using the center (h,k)=(3,−4)(h, k) = (3, -4) and the radius squared r2=25r^2 = 25.
(x−3)2+(y+4)2=25(x - 3)^2 + (y + 4)^2 = 25
The standard equation of a circle is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.
3
Expand the squared binomials in the standard equation to convert it into general form.
x2−6x+9+y2+8y+16=25x^2 - 6x + 9 + y^2 + 8y + 16 = 25
Expanding allows us to combine like terms and match the general form expressions in the options.
4
Simplify the expanded equation by combining constant terms and setting the equation to zero.
x2+y2−6x+8y+25=25⇒x2+y2−6x+8y=0x^2 + y^2 - 6x + 8y + 25 = 25 \Rightarrow x^2 + y^2 - 6x + 8y = 0
Subtracting 25 from both sides yields the final simplified general form equation of the circle.

Key Concept

Deriving and expanding the equation of a circle from its center and a point.
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Topics

Conic Sections

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