A circle is centered at $$ D(-1,3) $$. The point $$ G(-10,1) $$ is on the circle. Where does the point $$ J(-3,12) $$ lie?

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To determine where the point $$ J(-3,12) $$ lies in relation to the circle centered at $$ D(-1,3) $$ and passing through the point $$ G(-10,1) $$, we need to find the equation of the circle first.

The general equation of a circle with center $$ (h, k) $$ and radius $$ r $$ is given by:
$$ (x - h)^2 + (y - k)^2 = r^2 $$

Given that the circle is centered at $$ D(-1,3) $$, the equation of the circle is:
$$ (x + 1)^2 + (y - 3)^2 = r^2 $$

We also know that the point $$ G(-10,1) $$ lies on the circle, so we can substitute the coordinates of $$ G $$ into the equation to find the radius $$ r $$:
$$ (-10 + 1)^2 + (1 - 3)^2 = r^2 $$
$$ (-9)^2 + (-2)^2 = r^2 $$
$$ 81 + 4 = r^2 $$
$$ r^2 = 85 $$

Therefore, the equation of the circle is:
$$ (x + 1)^2 + (y - 3)^2 = 85 $$

Now, we need to determine where the point $$ J(-3,12) $$ lies in relation to this circle. We can substitute the coordinates of $$ J $$ into the equation of the circle to see if it satisfies the equation.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$\left(x+1\right)^{2}+\left(y-3\right)^{2}=85$$
- step1: Substitute:
  $$\left(-3+1\right)^{2}+\left(12-3\right)^{2}=85$$
- step2: Add the numbers:
  $$\left(-2\right)^{2}+\left(12-3\right)^{2}=85$$
- step3: Subtract the numbers:
  $$\left(-2\right)^{2}+9^{2}=85$$
- step4: Add the numbers:
  $$85=85$$
- step5: Check the equality:
  $$\textrm{true}$$
The point $$ J(-3,12) $$ lies on the circle centered at $$ D(-1,3) $$ and passing through the point $$ G(-10,1) $$.
The point $$ J(-3,12) $$ lies on the circle centered at $$ D(-1,3) $$ and passing through $$ G(-10,1) $$.

A circle is centered at . The point is on the circle.
Where does the point lie?

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