Standard Form of a Circle
The equation of a circle is written using the radius and center of the circle.
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The equation of the circle is shown with the center and radius of the circle. With this information, we can sketch the graph of the circle.
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Step by Step Guide to Write the Standard Form of a Circle
- The Standard form of a circle is \((x- h)^2+( y-k)^2= r^2\), where \(r\) is the radius and \((h, k)\) is the center. By knowing the center and radius of the circle we can write the standard form of a circle.
Standard form of a Circle – Example 1:
Write the standard form equation of circle with center: \((0, 5)\), radius: \(3\)
Solution:
The Standard form of a circle is \((x- h)^2+( y-k)^2= r^2\), where \(r\) is the radius and \((h, k)\) is the center.
In this case, the center is \((0, 5)\) and the radius is \(3\): \((h, k)=(0, 5), r=3\)
Then: \((x- 0)^2+( y-5)^2= 3^2 → x^2+( y-5)^2= 9 \)
Standard form of a Circle – Example 2:
Write the standard form equation of the circle \(x^2+y^2-6x+2y= 6\).
Solution:
The Standard form of a circle is \((x- h)^2+( y-k)^2= r^2\), where \(r\) is the radius and \((h, k)\) is the center.
Group \(x\)-variables and \(y\)-variables together: \((x^2-6x)+( y^2+2y)= 6\)
Convert \(x\) to square form: \((x^2-6x+9)+( y^2+2y)= 6+9 → (x-3)^2+( y^2+2y)=6+9\)
Convert \(y\) to square form: \((x-3)^2+( y^2+2y+1)= 6+9+1 → (x-3)^2+(y+1)^2=6+9+1\)
Then: \((x-3)^2+(y+1)^2=4^2\)
Exercises for Writing Standard form of a Circle
Write the standard form equation of each circle with the given information.
- \(\color{blue}{Center: (0, 4)}, \color{blue}{Radius: 2}\)
- \(\color{blue}{Center: (-1, 2)}\), \(\color{blue}{Radius: 5}\)
- \(\color{blue}{x^2+y^2-6x+8y=0}\)
- \(\color{blue}{x^2+y^2-2x+8y=0}\)
- \(\color{blue}{x^2+(y-4)^2=2^2}\)
- \(\color{blue}{(x+1)^2+(y-2)^2=5^2}\)
- \(\color{blue}{(x-5)^2+y^2=4^2}\)
- \(\color{blue}{(x-1)^2+(y+4)^2=5^2}\)
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