-2.4.27 find center and radius of circle

In summary, the graph of $x^2+y^2+6x+8y+9=0$ can be determined by completing the square to simplify the equation to $(x+3)^2+(y+4)^2=16=4^2$. This shows that the center of the graph is at $C(-3,-4)$ and the radius is $R=4$. While there may be other methods, completing the square is the most efficient technique.
  • #1
karush
Gold Member
MHB
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Determine the graph of $x^2+y^2+6x+8y+9=0$
$\begin{array}{rll}
\textsf{rewrite} &(x^2+6x )+(y^2+8y)=-9\\
\textsf{complete square} &(x^2+6x+9)+(y^2+8y+16)=-9+9+16\\
\textsf{simplify equation} &(x+3)^2+(y+4)^2=16=4^2\\
\textsf{observation} &C(-3,-4), \quad R=4
\end{array}$

hopefully ok
is there another way to do this other than complete the square

if you are inclined to do so I would be interested in a tikz code would be fore this :unsure:
 
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  • #2
this is ok

completing the square is probably the most efficient technique ...

Not saying there is no other method, but I'm not familiar with any.
 
  • #3
Mahalo
often when post here an alternative is suggested...
:cool:
 

1. What is the formula for finding the center and radius of a circle?

The formula for finding the center and radius of a circle is (h,k) for the center and r for the radius, where (h,k) is the coordinates of the center point and r is the distance from the center to any point on the circle.

2. How do you find the center and radius of a circle given its equation?

To find the center and radius of a circle given its equation, first rewrite the equation in the standard form (x-h)^2 + (y-k)^2 = r^2. Then, the coordinates of the center are (h,k) and the radius is the square root of r^2.

3. What information is needed to find the center and radius of a circle?

To find the center and radius of a circle, you need either the coordinates of the center point and the radius, or the equation of the circle in standard form.

4. Can you find the center and radius of a circle if only given three points on the circle?

Yes, you can find the center and radius of a circle if only given three points on the circle. First, find the equation of the perpendicular bisectors of two of the line segments connecting the three points. The intersection of these two bisectors is the center of the circle. Then, the distance from the center to any of the three points is the radius.

5. Is it possible for a circle to have a negative radius?

No, a circle cannot have a negative radius. The radius of a circle represents the distance from the center to any point on the circle, so it must be a positive value.

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