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?

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 \( G(-10,1) \).