Each point that lies on a circle satisfies the equation of that circle

What are some effective ways to find the equation of a circle when you are given points lying on the circle and the equation for the line on which the centre of the circle lies.

Here is an example of such a question:

Find the equation of a circle passing through the points $(4,1)$ and $(6,5)$ and whose centre lies on the line $4x + y = 16$

I have figured out one way to get the solution.

Let the $(h,k)$ be center of the circle, Since the line $4x + y = 16$ also includes the center $(h,k)$, then the following must be true,$$4h + k = 16\quad\quad\quad\text{(1)}$$

Let $r$ be the radius and $(x,y)$ be any point on the circle. Then, the equation satisfied by all points of the circle is $$(x-h)^2 + (y-k)^2 = r^2$$

Using this equation, $$(4-h)^2 - (1-k)^2 = r^2\quad\wedge\quad(6-h)^2 + (5-k)^2 = r^2$$ On equating and simplifying ,$$ h + 2k = 11 \quad\quad\quad\text{(2)}$$

On solving the linear equations $\text{(1) and (2)}$, $$ h = 3,\space k= 4 \quad\Rightarrow\quad \text{The centre is } (3,4)$$

Sticking that into an equation satisfied by a given point, $$ r^2 = (6-3)^2 + (5-4)^2 \quad\Rightarrow\quad r = \sqrt 10$$

I could have also used the distance formula ($d = \sqrt{(x_2 - x_1)^2 - (y_2 - y_1)^2}$) to find the radius.

But I want another approach. It feels too limiting just have only one or two ways to answer a question. I need have new perspectives. Please enlighten me with a different way to do this.

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In the standard

coordinate plane, what is the radius and the center of the circle

Possible Answers:

Correct answer:

Explanation:

When finding the center and radius of circle

, the center is

and the radius is

. Notice that they are not negative even though in the equation they have negative signs in front. This becomes important when dealing with real numbers. Also, notice the square of

Our circle,

has the same principles applied as the above principle, therefore

is our center. Notice how the numbers signs have been switched. This is the case for all circles due to the negative in the base equation above.

To find the radius of a circle, you must take the number the equation is equal to and square root it. This is due to the square of

mentioned above. The

. Use the least common multiples of 27 to find that three 3’s make up 27. Take two threes out as the square root of a number multiplied by itself is itself. This leaves one 3 under the radical. Therefore our radius is

Center:

Radius:

In the standard

coordinate plane, what is the area of the circle

Possible Answers:

Correct answer:

Explanation:

The general equation of a circle is

According to the question,

. Thus,

The general equation for the area of a circle is

When we plug in 13 for

, we get our area to equal

A circle in the standard coordinate plane is tangent to the x-axis at (3,0) and tangent to the y-axis at (0,3). What is the equation of the circle?

Possible Answers:

Correct answer:

Explanation:

The formula for the equation of a circle is (x – h)2+ (y – k)2 = r2, where (h, k) represents the coordinates of the center of the circle, and r represents the radius of the circle.

If a circle is tangent to the x-axis at (3,0), this means it touches the x-axis at that point. If a circle is tangent to the y-axis at (0,3), this means it touches the y-axis at that point. Given these two points, we can determine the center and the radius of the circle. The center of the circle must be equidistant from any of the points on the circumference. This means that both (0,3) and (3,0) are the same distance from the center. If we draw these points on a coodinate plane, it becomes apparent that the center of the circle must be (3,3). This point is exactly three units from each of the given points, indicating that the radius of the circle is 3.

When we input this information into the formula for a circle, we get (x – 3)2 + (y – 3)2 = 9.

Find the equation of the circle with center coordinates of

and a radius of

Possible Answers:

Correct answer:

Explanation:

The equation of a circle is

The center is

or, written another way

. Substituting

for

and

for

, our formula becomes

Finally, the formula of the circle is

On the xy plane, what is the area of a circle with the following equation:

Possible Answers:

Correct answer:

Explanation:

The standard form equation of a circle is

, where

is the center of the circle and

is equal to the radius. Thus, since we have the circle's standard form equation already given to us, we can ignore

and

, since all we need is

The area of circle is equal to

, which is equal to

A circle has a center at (5,5) and a radius of 2. If the format of the equation for the circle is (x-A)2+(y-B)2=C, what is C?

Possible Answers:

Explanation:

The circle has a center at (5,5) and a radius of 2. Therefore, the equation is (x-5)2+(y-5)2=22, or (x-5)2+(y-5)2=4.

If the center of a circle is at (0,4) and the diameter of the circle is 6, what is the equation of that circle?

Possible Answers:

Correct answer:

x2 + (y-4)2 = 9

Explanation:

The formula for the equation of a circle is:

(x-h) 2 + (y-k)2 = r2

Where (h,k) is the center of the circle.

h = 0 and k = 4

and diameter = 6 therefore radius = 3

(x-0) 2 + (y-4)2 = 32

x2 + (y-4)2 = 9

Circle A is given by the equation (x – 4)2 + (y + 3)2 = 29. Circle A is shifted up five units and left by six units. Then, its radius is doubled. What is the new equation for circle A?

Possible Answers:

(x – 10)2 + (y + 8)2 = 116

(x + 2)2 + (y – 2)2 = 116

(x – 10)2 + (y + 8)2 = 58

Correct answer:

(x + 2)2 + (y – 2)2 = 116

Explanation:

The general equation of a circle is (x – h)2 + (y – k)2 = r2, where (h, k) represents the location of the circle's center, and r represents the length of its radius.

Circle A first has the equation of (x – 4)2 + (y + 3)2 = 29. This means that its center must be located at (4, –3), and its radius is √29.

We are then told that circle A is shifted up five units and then left by six units. This means that the y-coordinate of the center would increase by five, and the x-coordinate of the center would decrease by 6. Thus, the new center would be located at (4 – 6, –3 + 5), or (–2, 2).

We are then told that the radius of circle A is doubled, which means its new radius is 2√29.

Now, that we have circle A's new center and radius, we can write its general equation using (x – h)2 + (y – k)2 = r2.

(x – (–2))2 + (y – 2)2 = (2√29)2 = 22(√29)2 = 4(29) = 116.

(x + 2)2 + (y – 2)2 = 116.

The answer is (x + 2)2 + (y – 2)2 = 116.

Which of the following equations describes all the points (x, y) in a coordinate plane that are five units away from the point (–3, 6)?

Possible Answers:

Correct answer:

(x + 3)2 + (y – 6)2 = 25

Explanation:

We are trying to find an equation for all of the points that are the same distance (5 units) from (–3, 6). The locus of all points equidistant from a single point is a circle. In other words, we need to find an equation of a circle. The center of the circle will be (–3, 6), and the radius, which is the distance from (–3,6), will be 5.

The standard form of a circle is given below:

(x – h)2 + (y – k)2 = r2, where the center is located at (h, k) and r is the length of the radius.

In this case, h will be –3, k will be 6, and r will be 5.

(x – (–3))2 + (y – 6)2 = 52

(x + 3)2 + (y – 6)2 = 25

The answer is (x + 3)2 + (y – 6)2 = 25.

What is the equation for a circle of radius 12, centered at the intersection of the two lines:

y1 = 4x + 3

and

y2 = 5x + 44?

Possible Answers:

(x - 22)2 + (y - 3)2 = 12

(x - 41)2 + (y - 161)2 = 144

None of the other answers

(x + 41)2 + (y + 161)2 = 144

(x - 3)2 + (y - 44)2 = 144

Correct answer:

(x + 41)2 + (y + 161)2 = 144

Explanation:

To begin, let us determine the point of intersection of these two lines by setting the equations equal to each other:

4x + 3 = 5x + 44; 3 = x + 44; –41 = x

To find the y-coordinate, substitute into one of the equations. Let's use y1:

y = 4 * –41 + 3 = –164 + 3 = –161

The center of our circle is therefore: (–41, –161).

Now, recall that the general form for a circle with center at (x0, y0) is:

(x - x0)2 + (y - y0)2 = r2

For our data, this means that our equation is:

(x + 41)2 + (y + 161)2 = 122 or (x + 41)2 + (y + 161)2 = 144

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