Question
Write general conic form equation using Center: (6, -1) Radius: 10
Answers
GPT-4o mini
The general form of the equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given the center \((6, -1)\) and the radius \(10\), we can substitute these values into the equation:
1. \(h = 6\)
2. \(k = -1\)
3. \(r = 10\)
Now substituting:
\[
(x - 6)^2 + (y + 1)^2 = 10^2
\]
Calculating \(10^2\):
\[
(x - 6)^2 + (y + 1)^2 = 100
\]
This is the equation of the circle in standard form. In general conic form (which typically takes the form \(Ax^2 + By^2 + Cx + Dy + E = 0\)), we can rearrange it:
\[
(x - 6)^2 + (y + 1)^2 - 100 = 0
\]
Expanding this equation gives us:
\[
(x^2 - 12x + 36) + (y^2 + 2y + 1) - 100 = 0
\]
Combining like terms:
\[
x^2 + y^2 - 12x + 2y - 63 = 0
\]
Thus, the general conic form of the equation is:
\[
x^2 + y^2 - 12x + 2y - 63 = 0
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given the center \((6, -1)\) and the radius \(10\), we can substitute these values into the equation:
1. \(h = 6\)
2. \(k = -1\)
3. \(r = 10\)
Now substituting:
\[
(x - 6)^2 + (y + 1)^2 = 10^2
\]
Calculating \(10^2\):
\[
(x - 6)^2 + (y + 1)^2 = 100
\]
This is the equation of the circle in standard form. In general conic form (which typically takes the form \(Ax^2 + By^2 + Cx + Dy + E = 0\)), we can rearrange it:
\[
(x - 6)^2 + (y + 1)^2 - 100 = 0
\]
Expanding this equation gives us:
\[
(x^2 - 12x + 36) + (y^2 + 2y + 1) - 100 = 0
\]
Combining like terms:
\[
x^2 + y^2 - 12x + 2y - 63 = 0
\]
Thus, the general conic form of the equation is:
\[
x^2 + y^2 - 12x + 2y - 63 = 0
\]