Find the Domain (x-1)^2+(y-4)^2=25

${(x - 1)}^{2} + {(y - 4)}^{2} = 25$

Step 1

Subtract ${(x - 1)}^{2}$ from both sides of the equation.

${(y - 4)}^{2} = 25 - {(x - 1)}^{2}$

Step 2

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

$y - 4 = \pm \sqrt{25 - {(x - 1)}^{2}}$

Step 3

Simplify $\pm \sqrt{25 - {(x - 1)}^{2}}$ .

Rewrite $25$ as $5^{2}$ .

$y - 4 = \pm \sqrt{5^{2} - {(x - 1)}^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x - 1$ .

$y - 4 = \pm \sqrt{(5 + x - 1) (5 - (x - 1))}$

Simplify.

Subtract $1$ from $5$ .

$y - 4 = \pm \sqrt{(x + 4) (5 - (x - 1))}$

Apply the distributive property.

$y - 4 = \pm \sqrt{(x + 4) (5 - x - - 1)}$

Multiply $- 1$ by $- 1$ .

$y - 4 = \pm \sqrt{(x + 4) (5 - x + 1)}$

Add $5$ and $1$ .

$y - 4 = \pm \sqrt{(x + 4) (- x + 6)}$

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.

$y - 4 = \sqrt{(x + 4) (- x + 6)}$

Add $4$ to both sides of the equation.

$y = \sqrt{(x + 4) (- x + 6)} + 4$

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

$y - 4 = - \sqrt{(x + 4) (- x + 6)}$

Add $4$ to both sides of the equation.

$y = - \sqrt{(x + 4) (- x + 6)} + 4$

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

$y = \sqrt{(x + 4) (- x + 6)} + 4$

$y = - \sqrt{(x + 4) (- x + 6)} + 4$

$y = \sqrt{(x + 4) (- x + 6)} + 4$

$y = - \sqrt{(x + 4) (- x + 6)} + 4$

Step 5

Set the radicand in $\sqrt{(x + 4) (- x + 6)}$ greater than or equal to $0$ to find where the expression is defined.

$(x + 4) (- x + 6) \geq 0$

Step 6

Solve for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 4 = 0$

$- x + 6 = 0$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Set $- x + 6$ equal to $0$ and solve for $x$ .

Set $- x + 6$ equal to $0$ .

$- x + 6 = 0$

Solve $- x + 6 = 0$ for $x$ .

Subtract $6$ from both sides of the equation.

$- x = - 6$

Divide each term in $- x = - 6$ by $- 1$ and simplify.

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x = 6$

The final solution is all the values that make $(x + 4) (- x + 6) \geq 0$ true.

$x = - 4, 6$

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 6$

$x > 6$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

$((- 6) + 4) (- (- 6) + 6) \geq 0$

The left side $- 24$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 4 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 6$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$((0) + 4) (- (0) + 6) \geq 0$

The left side $24$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$((8) + 4) (- (8) + 6) \geq 0$

The left side $- 24$ is less than the right side $0$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 4$ False

$- 4 < x < 6$ True

$x > 6$ False

$x < - 4$ False

$- 4 < x < 6$ True

$x > 6$ False

The solution consists of all of the true intervals.

$- 4 \leq x \leq 6$

Step 7

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 4, 6]$

Set-Builder Notation:

${x | - 4 \leq x \leq 6}$

Step 8

Find the Domain (x-1)^2+(y-4)^2=25

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