Question

Write general conic form equation of each circle
Center: (6, -1)
Radius: 10

Answer 1

The general conic form equation of a circle with center \((h, k)\) and radius \(r\) can be expressed as:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

For the given center \((6, -1)\) and radius \(10\):

- \(h = 6\)
- \(k = -1\)
- \(r = 10\)

Substituting these values into the equation:

\[
(x - 6)^2 + (y + 1)^2 = 10^2
\]

This simplifies to:

\[
(x - 6)^2 + (y + 1)^2 = 100
\]

This equation describes the circle in standard form.

If you want it in the general conic form, you can expand it:

1. Expand \((x - 6)^2\):
\[
(x - 6)^2 = x^2 - 12x + 36
\]

2. Expand \((y + 1)^2\):
\[
(y + 1)^2 = y^2 + 2y + 1
\]

Now, combine these:

\[
x^2 - 12x + 36 + y^2 + 2y + 1 = 100
\]

Combine like terms:

\[
x^2 + y^2 - 12x + 2y + 37 = 100
\]

To get it into the general conic form, subtract \(100\) from both sides:

\[
x^2 + y^2 - 12x + 2y - 63 = 0
\]

Therefore, the general conic form equation of the circle is:

\[
x^2 + y^2 - 12x + 2y - 63 = 0
\]