What is the standard form of the equation of a circle given by [tex]x^2+y^2-18 x+8 y+5=0[/tex]?
Answer (1)
Group the x and y terms and move the constant to the right side: x 2 − 18 x + y 2 + 8 y = − 5 .
Complete the square for x and y terms: ( x 2 − 18 x + 81 ) + ( y 2 + 8 y + 16 ) = − 5 + 81 + 16 .
Rewrite as binomial squares: ( x − 9 ) 2 + ( y + 4 ) 2 = 92 .
The standard form of the equation is: ( x − 9 ) 2 + ( y + 4 ) 2 = 92 .
Explanation
Understanding the Problem We are given the equation of a circle in general form: x 2 + y 2 − 18 x + 8 y + 5 = 0 . Our goal is to convert this equation into the standard form of a circle's equation, which is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) represents the center of the circle and r is the radius.
Grouping Terms First, we group the x and y terms together and move the constant term to the right side of the equation: x 2 − 18 x + y 2 + 8 y = − 5
Completing the Square Next, we complete the square for the x terms. To do this, we take half of the coefficient of the x term, which is − 18 , and square it: ( − 18/2 ) 2 = ( − 9 ) 2 = 81 . We add this value to both sides of the equation. Similarly, we complete the square for the y terms. We take half of the coefficient of the y term, which is 8 , and square it: ( 8/2 ) 2 = ( 4 ) 2 = 16 . We add this value to both sides of the equation. x 2 − 18 x + 81 + y 2 + 8 y + 16 = − 5 + 81 + 16
Rewriting as Binomial Squares Now, we rewrite the left side of the equation as the sum of two binomial squares: ( x − 9 ) 2 + ( y + 4 ) 2 = − 5 + 81 + 16
Simplifying the Right Side We simplify the right side of the equation: − 5 + 81 + 16 = 92
Standard Form Equation Thus, the equation in standard form is: ( x − 9 ) 2 + ( y + 4 ) 2 = 92 The center of the circle is ( 9 , − 4 ) and the radius is 92 .
Final Answer Therefore, the standard form of the equation of the circle is ( x − 9 ) 2 + ( y + 4 ) 2 = 92 .
Examples
Understanding the standard form of a circle's equation is useful in various real-world applications. For example, 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 location is (9, -4) and its range is defined by a radius of 92 units, the GPS can use the equation ( x − 9 ) 2 + ( y + 4 ) 2 = 92 to quickly determine if a device at location (x, y) is within the tower's coverage area. This concept extends to other scenarios like determining the coverage area of Wi-Fi routers or defining safe zones around hazardous areas.
