Writing a circle equation from center and radius

Key takeaway: A circle equation comes directly from the center-radius form. First find the radius from the center to the given point, then substitute into the standard equation.

Key idea: use the center-radius form

The standard form of a circle with center $(h,k)$ and radius $r$ is

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\$$ Here, the center is \$(-6,2)\$, and the point on the circle is \$(-9,-2)\$. That means we first compute the radius using the distance from the center to the point. ## Step 1: find the radius Use the distance formula: \ $$r = \sqrt{(-9-(-6))^2 + (-2-2)^2} \$$ \ $$r = \sqrt{(-3)^2 + (-4)^2} \$$ \ $$r = \sqrt{9+16} = \sqrt{25} = 5 \$$ ## Step 2: write the equation Substitute \$(h,k)=(-6,2)\$ and \$r=5\$: \ $$(x+6)^2 + (y-2)^2 = 25 \$$ That is the equation of the circle. ## Why this works Every point on the circle is exactly 5 units from the center \$(-6,2)\$. The equation encodes that fixed distance by setting the squared distance equal to \$25\$. ## Final answer \ $$\boxed{(x+6)^2 + (y-2)^2 = 25}\$$ --- **Pitfalls the pros know** 👇 A common mistake is writing the radius as 25 instead of 5. The radius is the distance, while 25 is the radius squared. Another issue is sign handling: the center \$(-6,2)\$ becomes \$x+6\$ and \$y-2\$, not \$x-6\$ or \$y+2\$. Also, the distance formula must use both coordinate differences, not just the x-values. If you skip the y-difference, you may still get the right-looking number by accident, but the reasoning will be incomplete. **What if the problem changes?** If the point on the circle were \$(-6,7)\$ instead of \$(-9,-2)\$, the radius would be the vertical distance from the center, so \$r=5\$ again and the equation would still be \$ (x+6)^2 + (y-2)^2 = 25 \$. If the center changed to \$(3,-1)\$ while the point became \$ (6,3) \$, then you would first find the new radius with the distance formula and then substitute into the same center-radius form. `Tags`: center-radius form, distance formula, radius of a circle