$x^2+y^2-14 x+12 y+69=0$ is the equation of a circle with center $(h, k)$ and radius $r$ for:
$h=$
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$k=$
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$r=$
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Answer (2)
Complete the square for the x terms: x 2 − 14 x becomes ( x − 7 ) 2 − 49 .
Complete the square for the y terms: y 2 + 12 y becomes ( y + 6 ) 2 − 36 .
Rewrite the original equation in standard form: ( x − 7 ) 2 + ( y + 6 ) 2 = 16 .
Identify the center and radius: The center is ( 7 , − 6 ) and the radius is 4 , so h = 7 , k = − 6 , r = 4 .
Explanation
Analyze the problem and rewrite the equation We are given the equation of a circle: x 2 + y 2 − 14 x + 12 y + 69 = 0 . Our goal is to find the center ( h , k ) and the radius r of this circle. To do this, we need to rewrite the equation in the standard form of a circle, which is ( x − h ) 2 + ( y − k ) 2 = r 2 . This involves completing the square for both the x and y terms.
Complete the square for x terms First, let's complete the square for the x terms. We have x 2 − 14 x . To complete the square, we take half of the coefficient of the x term, which is − 14/2 = − 7 , and square it: ( − 7 ) 2 = 49 . So, we can rewrite x 2 − 14 x as ( x − 7 ) 2 − 49 .
Complete the square for y terms Next, let's complete the square for the y terms. We have y 2 + 12 y . To complete the square, we take half of the coefficient of the y term, which is 12/2 = 6 , and square it: ( 6 ) 2 = 36 . So, we can rewrite y 2 + 12 y as ( y + 6 ) 2 − 36 .
Substitute back into the original equation Now, substitute these back into the original equation: ( x − 7 ) 2 − 49 + ( y + 6 ) 2 − 36 + 69 = 0 .
Simplify the equation Simplify the equation: ( x − 7 ) 2 + ( y + 6 ) 2 − 49 − 36 + 69 = 0 . Combine the constants: − 49 − 36 + 69 = − 16 . So, the equation becomes ( x − 7 ) 2 + ( y + 6 ) 2 − 16 = 0 .
Rewrite in standard form Add 16 to both sides to get the standard form: ( x − 7 ) 2 + ( y + 6 ) 2 = 16 .
Identify center and radius Now we can identify the center ( h , k ) and the radius r . Comparing the equation with the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , we have h = 7 , k = − 6 , and r 2 = 16 . Taking the square root of r 2 , we get r = 4 .
State the final answer Therefore, the center of the circle is ( 7 , − 6 ) and the radius is 4 . So, h = 7 , k = − 6 , and r = 4 .
Examples
Understanding the equation of a circle is crucial in various fields. For instance, in GPS technology, your location is determined by finding the intersection of circles from multiple satellites. Each satellite provides a distance (radius) from itself, defining a circle. The GPS receiver then solves for the center of these circles to pinpoint your exact coordinates on Earth. This principle relies heavily on the circle equation and its parameters.
The center of the circle is found at ( 7 , − 6 ) and its radius is 4 . This is determined by rewriting the given equation in standard form through completing the square for both the x and y terms. The final values are h = 7 , k = − 6 , and r = 4 .
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