Solve for r (x-h)^2+(y-k)^2=r^2

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

Step 1

Rewrite the equation as $r^{2} = {(x - h)}^{2} + {(y - k)}^{2}$ .

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

Step 2

Take the square root of both sides of the equation to eliminate the exponent on the left side.

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

Step 3

Simplify $\pm \sqrt{{(x - h)}^{2} + {(y - k)}^{2}}$ .

Rewrite ${(x - h)}^{2}$ as $(x - h) (x - h)$ .

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

Expand $(x - h) (x - h)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$r = \pm \sqrt{x \cdot x + x (- h) - h (x - h) + {(y - k)}^{2}}$

Apply the distributive property.

$r = \pm \sqrt{x \cdot x + x (- h) - h x - h (- h) + {(y - k)}^{2}}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

$r = \pm \sqrt{x^{2} - x h - h x - 1 \cdot - 1 h \cdot h + {(y - k)}^{2}}$

Multiply $h$ by $h$ by adding the exponents.

Move $h$ .

$r = \pm \sqrt{x^{2} - x h - h x - 1 \cdot - 1 (h \cdot h) + {(y - k)}^{2}}$

Multiply $h$ by $h$ .

$r = \pm \sqrt{x^{2} - x h - h x - 1 \cdot - 1 h^{2} + {(y - k)}^{2}}$

Multiply $- 1$ by $- 1$ .

$r = \pm \sqrt{x^{2} - x h - h x + 1 h^{2} + {(y - k)}^{2}}$

Multiply $h^{2}$ by $1$ .

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

Subtract $h x$ from $- x h$ .

Move $x$ .

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

Subtract $h x$ from $- h x$ .

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

Rewrite ${(y - k)}^{2}$ as $(y - k) (y - k)$ .

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

Expand $(y - k) (y - k)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y \cdot y + y (- k) - k (y - k)}$

Apply the distributive property.

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y \cdot y + y (- k) - k y - k (- k)}$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - y k - k y - 1 \cdot - 1 k \cdot k}$

Multiply $k$ by $k$ by adding the exponents.

Move $k$ .

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - y k - k y - 1 \cdot - 1 (k \cdot k)}$

Multiply $k$ by $k$ .

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - y k - k y - 1 \cdot - 1 k^{2}}$

Multiply $- 1$ by $- 1$ .

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - y k - k y + 1 k^{2}}$

Multiply $k^{2}$ by $1$ .

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - y k - k y + k^{2}}$

Subtract $k y$ from $- y k$ .

Move $y$ .

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - k y - k y + k^{2}}$

Subtract $k y$ from $- k y$ .

$r = \pm \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - 2 k y + k^{2}}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$r = \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - 2 k y + k^{2}}$

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - 2 k y + k^{2}}$

The complete solution is the result of both the positive and negative portions of the solution.

$r = \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - 2 k y + k^{2}}$

$r = - \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - 2 k y + k^{2}}$

$r = \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - 2 k y + k^{2}}$

$r = - \sqrt{x^{2} - 2 h x + h^{2} + y^{2} - 2 k y + k^{2}}$

Solve for r (x-h)^2+(y-k)^2=r^2

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