Select the correct answer.

Consider this equation of a circle.
[tex]x^2-2 x+y^2+6 y-4=0[/tex]

What is the equation expressed in standard form?
A. [tex](x-1)^2+(y+3)^2=14[/tex]
B. [tex](x-2)^2+(y+6)^2=44[/tex]
C. [tex](x-1)^2+(y+3)^2=56[/tex]
D. [tex](x-2)^2+(y+6)^2=12[/tex]

To convert the given circle equation to standard form:

Complete the square for x terms: x 2 − 2 x becomes ( x − 1 ) 2 − 1 .
Complete the square for y terms: y 2 + 6 y becomes ( y + 3 ) 2 − 9 .
Substitute back into the original equation and simplify: ( x − 1 ) 2 − 1 + ( y + 3 ) 2 − 9 − 4 = 0 .
Rearrange to standard form: ( x − 1 ) 2 + ( y + 3 ) 2 = 14 . The final answer is ( x − 1 ) 2 + ( y + 3 ) 2 = 14 ​ .

Explanation

Analyze the problem We are given the equation of a circle: x 2 − 2 x + y 2 + 6 y − 4 = 0 . Our goal is to rewrite this equation in the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius.

Complete the square To convert the given equation to standard form, we need to complete the square for both the x and y terms. For the x terms, we have x 2 − 2 x . To complete the square, we take half of the coefficient of the x term, which is − 2 , and square it: ( 2 − 2 ​ ) 2 = ( − 1 ) 2 = 1 . So, we can rewrite x 2 − 2 x as ( x − 1 ) 2 − 1 . For the y terms, we have y 2 + 6 y . To complete the square, we take half of the coefficient of the y term, which is 6 , and square it: ( 2 6 ​ ) 2 = ( 3 ) 2 = 9 . So, we can rewrite y 2 + 6 y as ( y + 3 ) 2 − 9 .

Substitute and simplify Now, substitute these expressions back into the original equation: ( x − 1 ) 2 − 1 + ( y + 3 ) 2 − 9 − 4 = 0 . Combine the constants: ( x − 1 ) 2 + ( y + 3 ) 2 − 1 − 9 − 4 = 0 . ( x − 1 ) 2 + ( y + 3 ) 2 − 14 = 0 .

Isolate squared terms Add 14 to both sides of the equation to isolate the squared terms: ( x − 1 ) 2 + ( y + 3 ) 2 = 14 . This is the equation of the circle in standard form.

Identify the correct option The equation in standard form is ( x − 1 ) 2 + ( y + 3 ) 2 = 14 . Comparing this to the given options, we see that it matches option A.

State the final answer The equation of the circle in standard form is ( x − 1 ) 2 + ( y + 3 ) 2 = 14 .


Examples
Understanding the standard form of a circle's equation is useful in various real-world applications. For instance, consider a GPS system that needs to determine if a location is within a certain range of a cell tower. If the cell tower's coverage area is modeled as a circle, the GPS can use the standard form equation to quickly calculate whether the location's coordinates satisfy the equation, thus determining if it's within range. The equation ( x − h ) 2 + ( y − k ) 2 = r 2 allows for easy computation of distances from the center ( h , k ) to any point ( x , y ) .

Answered by GinnyAnswer | 2025-07-03

The equation of the circle x 2 − 2 x + y 2 + 6 y − 4 = 0 can be rewritten as ( x − 1 ) 2 + ( y + 3 ) 2 = 14 by completing the square for both variables. The correct answer from the choices given is option A. This standard form indicates the center of the circle is at (1, -3) with a radius of 14 ​ .
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Answered by Anonymous | 2025-07-04