Practice 1 Solutions

For problems 1–3, use the given graph.

Note

Problems 1–3

The radius can be estimated to be a natural number. Be sure to pay attention to the scale of the given coordinate plane.

Find the equation of circle A.

$center : (- 4, 0) radius : 1 {(x - (- 4))}^{2} + {(y - 0)}^{2} = 1^{2}$

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

Find the equation of circle B.

$center : (4, - 2) radius : 3 {(x - 4)}^{2} + {(y - (- 2))}^{2} = 3^{2}$

${(x - 4)}^{2} + {(y + 2)}^{2} = 9$

Find the equation of circle C.

$center : (3, - 1) radius : 5 {(x - 3)}^{2} + {(y - (- 1))}^{2} = 5^{2}$

${(x - 3)}^{2} + {(y + 1)}^{2} = 25$

Write the equation of the circle in the form ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ with a center of $(5.43, - 3.1)$ and a radius of 2.3 units.

${(x - 5.43)}^{2} + {(y - (- 3.1))}^{2} = 2 . 3^{2}$

${(x - 5.43)}^{2} + {(y + 3.1)}^{2} = 5.29$

Write the equation of a circle with center $(- 2 \sqrt{5}, \sqrt{6})$ and radius of $\sqrt{13}$ units.

${(x - (- 2 \sqrt{5}))}^{2} + {(y - \sqrt{6})}^{2} = {(\sqrt{13})}^{2}$

${(x + 2 \sqrt{5})}^{2} + {(y - \sqrt{6})}^{2} = 13$

Name the domain and range of the circle in set builder notation given the equation: ${(x + 12)}^{2} + {(y - 14)}^{2} = 100$

$center : (- 12, 14) radius : 10 domain : - 12 \pm 10 range : 14 \pm 10$

$domain : \{x | x \in ℝ, - 22 \leq x \leq - 2\} range : \{y | y \in ℝ, 4 \leq y \leq 24\}$

Note

You may need to graph the circle to determine the domain or range. Recall, the radius is the same from the center to any point on the circle.

Q: What is the radius?

A: 10

Q: What is the center?

A: (–12, 14)

Q: How far is the center from any point on the circle?

A: 10

Name the domain and range of the circle in set builder notation given the equation: ${(x - 4.2)}^{2} + {(y - 8.5)}^{2} = 1.21$

$center : (4.2, 8.5) radius : 1.1 domain : 4.2 \pm 1.1 range : 8.5 \pm 1.1$

$domain : \{x | x \in ℝ, 3.1 \leq x \leq 5.3\} range : \{y | y \in ℝ, 7.4 \leq y \leq 9.6\}$

A circle has a center at $(- 4, 6)$ and an endpoint at $(1, - 6)$ . Find the equation of the circle and graph.

$(- 4, 6) (1, - 6) r = \sqrt{{(- 4 - 1)}^{2} + {(6 - (- 6))}^{2}} r = \sqrt{{(- 5)}^{2} + {(12)}^{2}} r = \sqrt{169} r = 13$

${(x + 4)}^{2} + {(y - 6)}^{2} = 169$

Determine the equation of the circle when the endpoints of the diameter are $(- 5, 2)$ and $(7, 6)$ . Graph.

$center : (\frac{- 5 + 7}{2}, \frac{2 + 6}{2}) = (1, 4) (1, 4), (7, 6) r = \sqrt{{(1 - 7)}^{2} + {(4 - 6)}^{2}} r = \sqrt{{(- 6)}^{2} + {(- 2)}^{2}} r = \sqrt{40} {(x - 1)}^{2} + {(y - 4)}^{2} = {(\sqrt{40})}^{2}$

${(x - 1)}^{2} + {(y - 4)}^{2} = 40$

Graph and find the equation of the circle with the domain of $\{x | x \in ℝ, - 3 \leq x \leq 7\}$ and the range of $\{y | y \in ℝ, - 7 \leq x \leq 3\}$ .

$Center : (2, - 2) radius : 5$

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

Graph and find the equation of the circle with the domain of $\{x | x \in ℝ, 4 \leq x \leq 9\}$ and the range is $\{y | y \in ℝ, - 1 \leq y \leq 4\}$ .

$center : (6.5, 1.5) radius : 2.5$

${(x - 6.5)}^{2} + {(y - 1.5)}^{2} = 6.25$

Write the equation of a circle in standard form: $2 x^{2} - 4 x + 2 y^{2} + 8 y = 8$

$2 (x^{2} - 2 x + {(- \frac{2}{2})}^{2}) + 2 (y^{2} + 4 y + {(\frac{4}{2})}^{2}) = 8 + 2 {(- \frac{2}{2})}^{2} + 2 {(\frac{4}{2})}^{2} 2 {(x - 1)}^{2} + 2 {(y + 2)}^{2} = 8 + 2 (1) + 2 (4) 2 {(x - 1)}^{2} + 2 {(y + 2)}^{2} = 18 {(x - 1)}^{2} + {(y + 2)}^{2} = 9$

${(x - 1)}^{2} + {(y + 2)}^{2} = 9$

Note

Because all terms were divisible by 2, all terms could have been divided by 2 as the first step.

Write the equation of the circle in standard form: $x^{2} + y^{2} - 11 x - 12 y = - 3$

$x^{2} - 11 x + y^{2} - 12 y = - 3 (x^{2} - 11 x + {(- \frac{11}{2})}^{2}) + (y^{2} - 12 y + {(- \frac{12}{2})}^{2}) = - 3 + {(- \frac{11}{2})}^{2} + {(- \frac{12}{2})}^{2} {(x - \frac{11}{2})}^{2} + {(y - 6)}^{2} = - \frac{12}{4} + \frac{121}{4} + \frac{144}{4}$

${(x - \frac{11}{2})}^{2} + {(y - 6)}^{2} = \frac{253}{4}$

The current transmission tower is centered at $(- 6, 3)$ and has a range of 4 miles. The transmission company wants to build a new tower at $(1, 5)$ to increase the signal to include the nearby town. Determine how much further the new transmission tower will reach.

Original: $r = 4$

New:

$r = \sqrt{{(- 6 - 1)}^{2} + {(3 - 5)}^{2}} r = \sqrt{{(- 7)}^{2} + {(- 2)}^{2}} r = \sqrt{53} \approx 7.28$

New – original: $7.28 - 4 = 3.28$

The new transmission tower will reach an additional 3.28 miles.

Graph: $x^{2} + {(y - 2)}^{2} = 25$

$center : (0, 2) radius : 5$

${(x - 4)}^{2} + {(y - 6)}^{2} = 1$

$center : (4, 6) radius : 1$