Write the equation of the circle in standard form. Then identify the center and radius of the circle.
[tex]x^2+y^2-10 x+8 y+37=0[/tex]
center: ( $\square$, $\square$ )
radius: $\square$
Answer (1)
Rewrite the given equation by grouping x and y terms: ( x 2 − 10 x ) + ( y 2 + 8 y ) = − 37 .
Complete the square for both x and y terms: ( x 2 − 10 x + 25 ) + ( y 2 + 8 y + 16 ) = − 37 + 25 + 16 .
Express the equation in standard form: ( x − 5 ) 2 + ( y + 4 ) 2 = 4 .
Identify the center ( 5 , − 4 ) and radius r = 2 : ( 5 , − 4 ) , 2 .
Explanation
Analyze the problem We are given the equation of a circle: x 2 + y 2 − 10 x + 8 y + 37 = 0 . Our goal is to rewrite this equation in the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) represents the center of the circle and r is the radius.
Group x and y terms To transform the given equation into standard form, we'll use a technique called completing the square. First, group the x terms and y terms together and move the constant to the right side of the equation: ( x 2 − 10 x ) + ( y 2 + 8 y ) = − 37
Complete the square for x terms Now, we complete the square for the x terms. Take half of the coefficient of the x term (-10), square it, and add it to both sides. Half of -10 is -5, and ( − 5 ) 2 = 25 . So we add 25 to both sides: ( x 2 − 10 x + 25 ) + ( y 2 + 8 y ) = − 37 + 25
Complete the square for y terms Next, we complete the square for the y terms. Take half of the coefficient of the y term (8), square it, and add it to both sides. Half of 8 is 4, and 4 2 = 16 . So we add 16 to both sides: ( x 2 − 10 x + 25 ) + ( y 2 + 8 y + 16 ) = − 37 + 25 + 16
Rewrite in standard form Now, rewrite the expressions in parentheses as squared terms and simplify the right side: ( x − 5 ) 2 + ( y + 4 ) 2 = − 37 + 25 + 16 ( x − 5 ) 2 + ( y + 4 ) 2 = 4
Identify center and radius The equation is now in standard form: ( x − 5 ) 2 + ( y + 4 ) 2 = 4 . From this, we can identify the center and radius. The center is ( h , k ) = ( 5 , − 4 ) , and the radius is r = 4 = 2 .
State the final answer Therefore, the equation of the circle in standard form is ( x − 5 ) 2 + ( y + 4 ) 2 = 4 , the center of the circle is ( 5 , − 4 ) , and the radius is 2.
Examples
Understanding the equation of a circle is very useful in many real-world applications. For example, civil engineers use it when designing circular structures like tunnels or roundabouts. Architects use it to design curved windows or domes. Also, in computer graphics, circles are fundamental for drawing and manipulating images. Knowing the center and radius allows precise placement and scaling of these circular elements in various designs and applications.
