Find the centre and radius of the circle (x − 5)² + (y + 3)² = 9.

Step-by-step solution

1. 1

   Write down the standard form. A circle of centre $(h,k)$ and radius $r$ has equation

   $(x-h)^{2}+(y-k)^{2}=r^{2}$

   Every bracket in the standard form contains a minus, which is the detail the next step turns on.

2. 2

   Match the x-bracket. $(x-5)^{2}$ matches $(x-h)^{2}$ with $h=5$ directly.

3. 3

   Match the y-bracket carefully. The given bracket is $(y+3)^{2}$, and to fit the pattern it must be rewritten as a subtraction:

   $(y+3)^{2}=\big(y-(-3)\big)^{2}\quad\Longrightarrow\quad k=-3$

   Reading $k=3$ here is the classic sign error.

4. 4

   Extract the radius. The right-hand side equals $r^{2}$, not $r$:

   $r^{2}=9\quad\Longrightarrow\quad r=\sqrt9=3$

   (Only the positive root is taken, since a radius is a length.)

5. 5

   State the answer.

   $\boxed{\text{centre }(5,-3),\quad \text{radius }3}$

6. 6

   Check with a point on the circle. Moving $3$ units right from the centre gives $(8,-3)$; substituting: $(8-5)^{2}+(-3+3)^{2}=9+0=9$ ✓. Moving $3$ units up gives $(5,0)$: $0+9=9$ ✓. Both lie on the circle, confirming centre and radius together.