Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A circle in the
xy-plane has its center on the line
x=3. If the point
(4,5) lies on the circle and the radius is
sqrt2, which of the following could be the center of the circle?
Choose 1 answer:
(A)
(3,3)
(C)
(3,4)
(C)
(3,5)
(D)
(3,7)

A circle in the $\newline$ xy-plane has its center on the line $\newline$ $x=3$ . If the point $\newline$ $(4,5)$ lies on the circle and the radius is $\newline$ $\sqrt{2}$ , which of the following could be the center of the circle? $\newline$ Choose $1$ answer: $\newline$ (A) $\newline$ $(3,3)$ $\newline$ (C) $\newline$ $(3,4)$ $\newline$ (C) $\newline$ $(3,5)$ $\newline$ (D) $\newline$ $(3,7)$

View step-by-step help

View step-by-step help

Find trigonometric ratios using the unit circle

Full solution

Q. A circle in the

\newline

xy-plane has its center on the line

\newline

x=3

. If the point

\newline

(4,5)

lies on the circle and the radius is

\newline

\sqrt{2}

, which of the following could be the center of the circle?

\newline

Choose

1

answer:

\newline

(A)

\newline

(3,3)

\newline

(C)

\newline

(3,4)

\newline

(C)

\newline

(3,5)

\newline

(D)

\newline

(3,7)

Circle Distance Formula: Determine the distance formula for a circle with center $(h,k)$ and a point on the circle $(x,y)$ . The distance formula is given by the equation $(x-h)^2 + (y-k)^2 = r^2$ , where $r$ is the radius of the circle.
Plug in Values: Plug in the known values from the problem into the distance formula. We know the radius $r = \sqrt{2}$ , and the point on the circle is $(4,5)$ . The center of the circle has the form $(3,k)$ since it lies on the line $x=3$ . The distance formula becomes $(4-3)^2 + (5-k)^2 = (\sqrt{2})^2$ .
Simplify Equation: Simplify the equation from the previous step. This gives us $(1)^2 + (5-k)^2 = 2$ .
Find Value of $\newline$ $k$ : Further simplify the equation to find the value of $\newline$ $k$ . We have $\newline$ $1 + (5-k)^2 = 2$ . Subtract $\newline$ $1$ from both sides to get $\newline$ $(5-k)^2 = 1$ .
Solve for $k$ : Solve for $k$ . Since $(5-k)^2 = 1$ , we take the square root of both sides to get $5-k = \pm 1$ .
Possible Values for $k$ : Find the two possible values for $k$ . We have $k = 5 - 1$ or $k = 5 + 1$ , which gives us $k = 4$ or $k = 6$ .
Check Answer Choices: Check which of the possible values for $k$ corresponds to the answer choices. We have determined that $k$ could be $4$ or $6$ , but since $6$ is not an option in the answer choices, we conclude that the center of the circle is $(3,4)$ .

More problems from Find trigonometric ratios using the unit circle

Question

Find all solutions with

0^\circ \leq \theta \leq 180^\circ

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

\newline

\cos(\theta) = -1

\newline

\underline{\hspace{2em}}^\circ

Posted 1 month ago

Question

Kimi's school is

15

kilometers west of her house and

8

kilometers south of her friend Franklin's house. Every day, Kimi bicycles from her house to her school. After school, she bicycles from her school to Franklin's house. Before dinner, she bicycles home on a bike path that goes straight from Franklin's house to her own house. How far does Kimi bicycle each day?

\newline

\_\_\_\_\_

Posted 1 month ago

Question

Evaluate. Write your answer as an integer or as a decimal rounded to the nearest hundredth.

\newline

\cos 35^\circ =

Posted 1 month ago

Question

Find all solutions with

-90^\circ \leq \theta \leq 90^\circ

. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.

\newline

\csc(\theta) = -1

\newline

\_\_\_\_\,^\circ

Posted 1 month ago

Question

Evaluate. Write your answer in simplified, rationalized form. Do not round.

\newline

\cot 30^\circ =

Posted 1 month ago

Question

The terminal side of an angle

\theta

in standard position intersects the unit circle at

(\frac{84}{85}, \frac{13}{85})

\cos(\theta)

\newline

Write your answer in simplified, rationalized form.

\newline

Posted 1 month ago

Question

-90^\circ < \theta < 90^\circ

. Find the value of

\theta

\newline

\tan(\theta) = 0

\newline

Write your answer in simplified, rationalized form. Do not round.

\newline

\theta =

^\circ

\newline

Posted 2 months ago

Question

Evaluate. Write your answer in simplified, rationalized form. Do not round.

\newline

\csc 60^\circ =

Posted 1 month ago

Question

\theta

is an acute angle. Find the value of

\theta

\newline

\sec(\theta) = \sqrt{2}

\newline

\theta =

\degree

Posted 1 month ago

Question

Evaluate. Write your answer in simplified, rationalized form. Do not round.

\newline

\cos 90^\circ =

Posted 2 months ago