act Math assist » Algebra » Coordinate airplane » one » how to discover the equation the a one

In the standard 

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


*


*


*


*


Explanation:

When detect the center and radius of one

*
, the facility is 
*
and the radius is
*
. Notification that they are not an adverse even though in the equation lock have an unfavorable signs in front. This becomes necessary when handling real numbers. Also, notification the square the
*
.

You are watching: Equation of a circle centered at origin

Our circle, 

*
has the very same principles used as the over principle, therefore 
*
is ours center. An alert how the number signs have actually been switched. This is the situation for all circles as result of the an unfavorable in the basic equation above.

To find the radius of a circle, you have to take the number the equation is same to and also square root it. This is because of the square of 

*
stated above. The
*
. Usage the least typical multiples of 27 to uncover that 3 3’s make up 27. Take 2 threes out together the square source of a number multiply by itself is itself. This pipeline one 3 under the radical. As such our radius is
*
.

Center: 

*
Radius:
*


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Example question #2 : how To discover The Equation the A circle


In the standard 

*
coordinate plane, what is the area that the circle 
*
?


Possible Answers:
Correct answer:

*


Explanation:

The basic equation of a one is 

*
.

According to the question,

*
. Thus,
*
.

The general equation for the area the a one is

*
.

When we plug in 13 because that

*
, we acquire our area to equal
*
.


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Example question #3 : exactly how To uncover The Equation of A circle


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


Possible Answers:
Correct answer:

*


Explanation:

The formula for the equation that a one is (x – h)2+ (y – k)2 = r2, wherein (h, k) represents the works with of the center of the circle, and also r to represent the radius that the circle.

If a one is tangent come the x-axis in ~ (3,0), this way it touches the x-axis at the point. If a circle is tangent come the y-axis at (0,3), this way it touch the y-axis at the point. Offered these two points, we deserve to determine the center and also the radius that the circle. The facility of the circle have to be equidistant from any kind of of the point out on the circumference. This method that both (0,3) and (3,0) space the exact same distance indigenous the center. If we draw these clues on a coodinate plane, the becomes noticeable that the center of the circle have to be (3,3). This point is specifically three devices from each of the given points, indicating the the radius of the one is 3.

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


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Example concern #4 : exactly how To uncover The Equation that A circle


Find the equation the the one with facility coordinates the

*
 and a radius that
*
.


Possible Answers:
Correct answer:

*


Explanation:

The equation the a circle is

*

The facility is

*
 or, written another way
*
. Substituting
*
 for
*
 and
*
 for
*
, our formula becomes
*

Finally, the formula of the one is 

*


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Example question #5 : exactly how To uncover The Equation of A one


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

*


Possible Answers:
Correct answer:

*


Explanation:

The standard form equation that a circle is 

*
, where
*
 is the facility of the circle and
*
 is same to the radius. Thus, since we have actually the circle"s standard form equation already given come us, we deserve to ignore
*
 and
*
, since all we require is
*
.

The area of one is same to 

*
, which is same to
*
.


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Example concern #6 : just how To uncover The Equation the A one


A circle has actually a center at (5,5) and also a radius that 2. If the layout of the equation because that the one is (x+A)2+(y+B)2=C, what is C?


Possible Answers:

5


4


2


3


1


Correct answer:

4


Explanation:

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


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Example concern #7 : just how To uncover The Equation of A circle


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


Possible Answers:

x2 + (y-4)2 = 9


x2 + y2 = 9


x2 + (y-4)2 = 36


(x-4)2 + y2 = 36


(x-4)2 + y2 = 9


Correct answer:

x2 + (y-4)2 = 9


Explanation:

The formula for the equation of a one is:

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

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

h = 0 and k = 4

and diameter = 6 as such radius = 3

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

x2 + (y-4)2 = 9


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Example concern #8 : how To find The Equation of A circle


Circle A is offered by the equation (x – 4)2 + (y + 3)2 = 29. One 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 – 2)2 + (y + 2)2 = 58


(x – 10)2 + (y + 8)2 = 116


(x + 2)2 + (y – 2)2 = 58


(x – 10)2 + (y + 8)2 = 58


(x + 2)2 + (y – 2)2 = 116


Correct answer:

(x + 2)2 + (y – 2)2 = 116


Explanation:

The general equation the a one is (x – h)2 + (y – k)2 = r2, whereby (h, k) to represent the ar of the circle"s center, and also r to represent the length of its radius. 

Circle A an initial has the equation of (x – 4)2 + (y + 3)2 = 29. This way that its center must be situated at (4, –3), and also its radius is √29.

We room then told the circle A is shifted up 5 units and then left by 6 units. This method that the y-coordinate that the facility would boost by five, and also the x-coordinate that the center would diminish by 6. Thus, the new center would be situated at (4 – 6, –3 + 5), or (–2, 2).

We room then told that the radius of one A is doubled, which way its new radius is 2√29.

Now, the we have circle A"s new center and also 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 price is (x + 2)2 + (y – 2)2 = 116.


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Example question #9 : just how To find The Equation of A circle


Which that the following equations explains all the point out (x, y) in a coordinate plane that are five units far from the suggest (–3, 6)?


Possible Answers:

(x – 3)2 + (y + 6)2 = 5


(x + 3)2 + (y – 6)2 = 25


y + 6 = 5 – (x – 3)2


(x – 3)2 + (y + 6)2 = 25


(x – 3)2 – (y + 6)2 = 25


Correct answer:

(x + 3)2 + (y – 6)2 = 25


Explanation:

We room trying to uncover an equation for all of the points that room the same distance (5 units) native (–3, 6). The locus of every points equidistant indigenous a solitary point is a circle. In other words, we need to find an equation the a circle. The facility of the circle will certainly be (–3, 6), and the radius, i beg your pardon is the street from (–3,6), will certainly be 5. 

The standard type of a one is offered below:

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

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

(x – (–3))2 + (y – 6)2 = 52

(x + 3)2 + (y – 6)2 = 25

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

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Example question #1 : circles


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

y1 = 4x + 3

and

y2 = 5x + 44?


Possible Answers:

None that the various other answers


(x + 41)2 + (y + 161)2 = 144


(x - 22)2 + (y - 3)2 = 12


(x - 3)2 + (y - 44)2 = 144


(x - 41)2 + (y - 161)2 = 144


Correct answer:

(x + 41)2 + (y + 161)2 = 144


Explanation:

To begin, permit us identify the point of intersection that these two lines by setup the equations equal to each other:

4x + 3 = 5x + 44; 3 = x + 44; –41 = x

To discover 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 the the general form for a circle with facility at (x0, y0) is:

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

For our data, this method that our equation is:

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


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