Determine the equation of a circle that passes through the point 𝐴: zero, eight, if its center is 𝑀: negative two, negative six.
The general equation of the circle is given by 𝑥 minus 𝑎 all squared plus 𝑦 minus 𝑏 all squared equals 𝑟 squared, where the coordinates of the center of the circle is a, 𝑏 and the radius is 𝑟. In our example, 𝑎 is equal to negative two. And 𝑏 is equal to negative six. The radius of the circle, in this case, will be the distance from 𝐴 to 𝑀. This distance can be calculated using the formula: the square root of 𝑥 two minus 𝑥 one all squared plus 𝑦 two minus 𝑦 one all squared.
Substituting in the coordinates of 𝐴 and 𝑀 gives us square root of negative two minus zero all squared plus negative six minus eight all squared. This simplifies to negative two squared plus negative 14 squared. Negative two squared is four. And negative 14 squared is 196. Therefore, the length 𝐴𝑀, the radius of the circle, is the square root of 200.
We now need to substitute the coordinates of the center 𝑀 — negative two, negative six — and the radius, root 200, into the equation of a circle. This gives us 𝑥 minus negative two all squared plus 𝑦 minus negative six all squared equals root 200 squared. The left-hand side can be simplified to 𝑥 plus two all squared plus 𝑦 plus six all squared. On the right-hand side, the square root of 200 squared is 200.
This means that the equation of a circle that passes through the point zero, eight with a center negative two, negative six is: 𝑥 plus two all squared plus 𝑦 plus six all squared equals 200.