Graph the circle. [tex]x^2+y^2+2 x-4 y-11=0[/tex]

Rewrite the given equation x 2 + y 2 + 2 x − 4 y − 11 = 0 by completing the square for both x and y terms.
Complete the square for x : x 2 + 2 x = ( x + 1 ) 2 − 1 and for y : y 2 − 4 y = ( y − 2 ) 2 − 4 .
Substitute these back into the original equation and simplify to get the standard form: ( x + 1 ) 2 + ( y − 2 ) 2 = 16 .
Identify the center ( − 1 , 2 ) and radius 4 and graph the circle. Center: ( − 1 , 2 ) , Radius: 4 ​

Explanation

Analyze the problem and rewrite the equation The equation of the circle is given as x 2 + y 2 + 2 x − 4 y − 11 = 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. This will allow us to easily graph the circle.

Complete the square for x terms First, let's complete the square for the x terms. We have x 2 + 2 x . To complete the square, we need to add and subtract ( 2/2 ) 2 = 1 . So, x 2 + 2 x = ( x 2 + 2 x + 1 ) − 1 = ( x + 1 ) 2 − 1 .

Complete the square for y terms Next, let's complete the square for the y terms. We have y 2 − 4 y . To complete the square, we need to add and subtract ( − 4/2 ) 2 = 4 . So, y 2 − 4 y = ( y 2 − 4 y + 4 ) − 4 = ( y − 2 ) 2 − 4 .

Substitute back into the original equation Now, substitute these back into the original equation: x 2 + y 2 + 2 x − 4 y − 11 = 0 becomes (( x + 1 ) 2 − 1 ) + (( y − 2 ) 2 − 4 ) − 11 = 0 .

Simplify the equation to standard form Simplify the equation: ( x + 1 ) 2 − 1 + ( y − 2 ) 2 − 4 − 11 = 0 ( x + 1 ) 2 + ( y − 2 ) 2 = 1 + 4 + 11 ( x + 1 ) 2 + ( y − 2 ) 2 = 16 .

Identify the center and radius Now we can identify the center and radius of the circle. The equation is in the form ( x − h ) 2 + ( y − k ) 2 = r 2 , where h = − 1 , k = 2 , and r 2 = 16 . Therefore, the center of the circle is ( − 1 , 2 ) and the radius is r = 16 ​ = 4 .

Graph the circle To graph the circle, plot the center ( − 1 , 2 ) on the coordinate plane. Then, draw a circle with a radius of 4 units around this center.

State the final answer The equation of the circle is ( x + 1 ) 2 + ( y − 2 ) 2 = 16 . The center of the circle is ( − 1 , 2 ) and the radius is 4.


Examples
Circles are fundamental in many real-world applications, from designing gears and wheels in mechanical engineering to understanding the orbits of planets in astronomy. In architecture, arches and domes often incorporate circular shapes for their structural integrity and aesthetic appeal. Even in everyday life, understanding circles helps us calculate areas, perimeters, and volumes of cylindrical objects, which is useful in cooking, construction, and many other practical scenarios.

Answered by GinnyAnswer | 2025-07-03

The standard form of the circle is ( x + 1 ) 2 + ( y − 2 ) 2 = 16 , with a center at ( − 1 , 2 ) and a radius of 4. To graph, plot the center and draw a circle around it with a radius of 4 units. This provides a clear representation of the circle on a coordinate plane.
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Answered by Anonymous | 2025-07-04