Find the Circle Using the Diameter End Points (-2,6) , (5,1)

$(- 2, 6)$ , $(5, 1)$

Step 1

The diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circumference of the circle. The given end points of the diameter are $(- 2, 6)$ and $(5, 1)$ . The center point of the circle is the center of the diameter, which is the midpoint between $(- 2, 6)$ and $(5, 1)$ . In this case the midpoint is $(\frac{3}{2}, \frac{7}{2})$ .

Use the midpoint formula to find the midpoint of the line segment.

$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

Substitute in the values for $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

$(\frac{- 2 + 5}{2}, \frac{6 + 1}{2})$

Add $- 2$ and $5$ .

$(\frac{3}{2}, \frac{6 + 1}{2})$

Add $6$ and $1$ .

$(\frac{3}{2}, \frac{7}{2})$

Step 2

Find the radius $r$ for the circle. The radius is any line segment from the center of the circle to any point on its circumference. In this case, $r$ is the distance between $(\frac{3}{2}, \frac{7}{2})$ and $(- 2, 6)$ .

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Substitute the actual values of the points into the distance formula.

$r = \sqrt{{((- 2) - \frac{3}{2})}^{2} + {(6 - \frac{7}{2})}^{2}}$

Simplify.

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$r = \sqrt{{(- 2 \cdot \frac{2}{2} - \frac{3}{2})}^{2} + {(6 - \frac{7}{2})}^{2}}$

Combine $- 2$ and $\frac{2}{2}$ .

$r = \sqrt{{(\frac{- 2 \cdot 2}{2} - \frac{3}{2})}^{2} + {(6 - \frac{7}{2})}^{2}}$

Combine the numerators over the common denominator.

$r = \sqrt{{(\frac{- 2 \cdot 2 - 3}{2})}^{2} + {(6 - \frac{7}{2})}^{2}}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$r = \sqrt{{(\frac{- 4 - 3}{2})}^{2} + {(6 - \frac{7}{2})}^{2}}$

Subtract $3$ from $- 4$ .

$r = \sqrt{{(\frac{- 7}{2})}^{2} + {(6 - \frac{7}{2})}^{2}}$

Move the negative in front of the fraction.

$r = \sqrt{{(- \frac{7}{2})}^{2} + {(6 - \frac{7}{2})}^{2}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7}{2}$ .

$r = \sqrt{{(- 1)}^{2} {(\frac{7}{2})}^{2} + {(6 - \frac{7}{2})}^{2}}$

Apply the product rule to $\frac{7}{2}$ .

$r = \sqrt{{(- 1)}^{2} (\frac{7^{2}}{2^{2}}) + {(6 - \frac{7}{2})}^{2}}$

Raise $- 1$ to the power of $2$ .

$r = \sqrt{1 (\frac{7^{2}}{2^{2}}) + {(6 - \frac{7}{2})}^{2}}$

Multiply $\frac{7^{2}}{2^{2}}$ by $1$ .

$r = \sqrt{\frac{7^{2}}{2^{2}} + {(6 - \frac{7}{2})}^{2}}$

Raise $7$ to the power of $2$ .

$r = \sqrt{\frac{49}{2^{2}} + {(6 - \frac{7}{2})}^{2}}$

Raise $2$ to the power of $2$ .

$r = \sqrt{\frac{49}{4} + {(6 - \frac{7}{2})}^{2}}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$r = \sqrt{\frac{49}{4} + {(6 \cdot \frac{2}{2} - \frac{7}{2})}^{2}}$

Combine $6$ and $\frac{2}{2}$ .

$r = \sqrt{\frac{49}{4} + {(\frac{6 \cdot 2}{2} - \frac{7}{2})}^{2}}$

Combine the numerators over the common denominator.

$r = \sqrt{\frac{49}{4} + {(\frac{6 \cdot 2 - 7}{2})}^{2}}$

Simplify the numerator.

Multiply $6$ by $2$ .

$r = \sqrt{\frac{49}{4} + {(\frac{12 - 7}{2})}^{2}}$

Subtract $7$ from $12$ .

$r = \sqrt{\frac{49}{4} + {(\frac{5}{2})}^{2}}$

Simplify the expression.

Apply the product rule to $\frac{5}{2}$ .

$r = \sqrt{\frac{49}{4} + \frac{5^{2}}{2^{2}}}$

Raise $5$ to the power of $2$ .

$r = \sqrt{\frac{49}{4} + \frac{25}{2^{2}}}$

Raise $2$ to the power of $2$ .

$r = \sqrt{\frac{49}{4} + \frac{25}{4}}$

Combine the numerators over the common denominator.

$r = \sqrt{\frac{49 + 25}{4}}$

Add $49$ and $25$ .

$r = \sqrt{\frac{74}{4}}$

Cancel the common factor of $74$ and $4$ .

Factor $2$ out of $74$ .

$r = \sqrt{\frac{2 (37)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$r = \sqrt{\frac{2 \cdot 37}{2 \cdot 2}}$

Cancel the common factor.

$r = \sqrt{\frac{2 \cdot 37}{2 \cdot 2}}$

Rewrite the expression.

$r = \sqrt{\frac{37}{2}}$

Rewrite $\sqrt{\frac{37}{2}}$ as $\frac{\sqrt{37}}{\sqrt{2}}$ .

$r = \frac{\sqrt{37}}{\sqrt{2}}$

Multiply $\frac{\sqrt{37}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$r = \frac{\sqrt{37}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{37}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$r = \frac{\sqrt{37} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$r = \frac{\sqrt{37} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$r = \frac{\sqrt{37} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \frac{\sqrt{37} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \frac{\sqrt{37} \sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$r = \frac{\sqrt{37} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = \frac{\sqrt{37} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$r = \frac{\sqrt{37} \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \frac{\sqrt{37} \sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$r = \frac{\sqrt{37} \sqrt{2}}{2}$

Evaluate the exponent.

$r = \frac{\sqrt{37} \sqrt{2}}{2}$

Simplify the numerator.

Combine using the product rule for radicals.

$r = \frac{\sqrt{37 \cdot 2}}{2}$

Multiply $37$ by $2$ .

$r = \frac{\sqrt{74}}{2}$

Step 3

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ is the equation form for a circle with $r$ radius and $(h, k)$ as the center point. In this case, $r = \frac{\sqrt{74}}{2}$ and the center point is $(\frac{3}{2}, \frac{7}{2})$ . The equation for the circle is ${(x - (\frac{3}{2}))}^{2} + {(y - (\frac{7}{2}))}^{2} = {(\frac{\sqrt{74}}{2})}^{2}$ .

${(x - (\frac{3}{2}))}^{2} + {(y - (\frac{7}{2}))}^{2} = {(\frac{\sqrt{74}}{2})}^{2}$

Step 4

The circle equation is ${(x - \frac{3}{2})}^{2} + {(y - \frac{7}{2})}^{2} = \frac{37}{2}$ .

${(x - \frac{3}{2})}^{2} + {(y - \frac{7}{2})}^{2} = \frac{37}{2}$

Step 5

Find the Circle Using the Diameter End Points (-2,6) , (5,1)

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