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Difficulty: EasyConic Sections

A circle in the standard (x,y)(x, y) coordinate plane has its center at (5,−2)(5, -2) and a radius of 77 units. Which of the following is an equation of this circle?

  1. x2+y2−10x+4y=20x^2 + y^2 - 10x + 4y = 20Answer
  2. B
    x2+y2+10x−4y=20x^2 + y^2 + 10x - 4y = 20
  3. C
    x2+y2−10x+4y=−22x^2 + y^2 - 10x + 4y = -22
  4. D
    x2+y2=70x^2 + y^2 = 70
  5. E
    x2+y2−10x+4y=7x^2 + y^2 - 10x + 4y = 7

Answer

The equation x2+y2−10x+4y=20x^2 + y^2 - 10x + 4y = 20
The standard form 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. Substituting h=5h = 5, k=−2k = -2, and r=7r = 7 yields (x−5)2+(y+2)2=49(x - 5)^2 + (y + 2)^2 = 49. Expanding both binomials gives x2−10x+25+y2+4y+4=49x^2 - 10x + 25 + y^2 + 4y + 4 = 49. Combining the constant terms on the left side gives x2+y2−10x+4y+29=49x^2 + y^2 - 10x + 4y + 29 = 49. Subtracting 2929 from both sides results in the equation x2+y2−10x+4y=20x^2 + y^2 - 10x + 4y = 20.

Step-by-Step Solution

1
Write down the standard equation of a circle.
(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.
This is the fundamental formula relating the geometric properties of a circle to its algebraic representation.
2
Substitute the given values into the standard equation.
(x−5)2+(y−(−2))2=72(x - 5)^2 + (y - (-2))^2 = 7^2, which simplifies to (x−5)2+(y+2)2=49(x - 5)^2 + (y + 2)^2 = 49.
Substituting the center (5,−2)(5, -2) and radius 77 into the standard equation sets up the expression for expansion.
3
Expand the squared binomials.
x2−10x+25+y2+4y+4=49x^2 - 10x + 25 + y^2 + 4y + 4 = 49.
Expanding (x−5)2(x - 5)^2 into x2−10x+25x^2 - 10x + 25 and (y+2)2(y + 2)^2 into y2+4y+4y^2 + 4y + 4 allows conversion to the general form.
4
Simplify and rearrange the equation to match the form of the options.
x2+y2−10x+4y+29=49x^2 + y^2 - 10x + 4y + 29 = 49, which simplifies to x2+y2−10x+4y=20x^2 + y^2 - 10x + 4y = 20.
Combining the constants and subtracting 2929 from both sides of the equation yields the final simplified general form.

Key Concept

Equation of a Circle in General Form
Estimated Time:1m 0s
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Topics

Conic Sections

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