The circle given by $x^2+y^2-6 y-12=0$ can be written in standard form like this: $x^2+(y-k)^2=21$.
What is the value of $k$ in this equation?
Answer (1)
Rewrite the given equation by adding 12 to both sides: x 2 + y 2 − 6 y = 12 .
Complete the square for the y terms by adding ( 2 − 6 ) 2 = 9 to both sides: x 2 + y 2 − 6 y + 9 = 12 + 9 .
Rewrite the equation in standard form: x 2 + ( y − 3 ) 2 = 21 .
Identify the value of k by comparing with the standard form x 2 + ( y − k ) 2 = 21 : k = 3 .
Explanation
Analyze the problem We are given the equation of a circle: x 2 + y 2 − 6 y − 12 = 0 . We want to rewrite this equation in the standard form x 2 + ( y − k ) 2 = 21 and find the value of k .
Isolate x and y terms First, we add 12 to both sides of the given equation to isolate the terms involving x and y :
x 2 + y 2 − 6 y = 12
Complete the square Next, we complete the square for the y terms. To do this, we take half of the coefficient of the y term (which is -6), square it, and add it to both sides of the equation. Half of -6 is -3, and ( − 3 ) 2 = 9 . So, we add 9 to both sides: x 2 + y 2 − 6 y + 9 = 12 + 9 x 2 + ( y − 3 ) 2 = 21
Find the value of k Now, we compare the equation x 2 + ( y − 3 ) 2 = 21 with the standard form x 2 + ( y − k ) 2 = 21 . We can see that k = 3 .
State the final answer Therefore, the value of k in the equation x 2 + ( y − k ) 2 = 21 is 3.
Examples
Understanding the standard form of a circle's equation is useful in various real-world applications. For example, if you're designing a circular garden with a sprinkler at the center, knowing the equation of the circle helps you determine the garden's boundaries and the sprinkler's coverage area. By adjusting the parameters in the equation, you can easily modify the size and position of the garden to fit your available space and watering needs. This ensures efficient use of resources and optimal garden design.
