Equation of a Circle - ACT Math Test

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Question

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?

Answer

The formula for the equation of a circle is:

$(x-h)^{2}+(y-k)^{2}=r^{2}$

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

and

and diameter = 6 therefore radius = 3

$(x-0)^{2}+(y-4)^{2}=3^{2}$

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

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Question

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?

Answer

The general equation of a circle is $(x-h)^{2}+(y-k)^{2}=r^{2}$ , 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 $\sqrt{29}$ .

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 $2\sqrt{29}$ .

Now, that we have circle A's new center and radius, we can write its general equation using $(x-h)^{2}+(y-k)^{2}=r^{2}$ .

$(x-(-2))^{2}+(y-2)^{2}=(2\sqrt{29})^{2}=2^{2}(\sqrt{29})^{2}=4(29)=116$ .

$(x+2)^{2}+(y-2)^{2}=116$ .

The answer is $(x+2)^{2}+(y-2)^{2}=116$ .

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Question

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

$y_{1}=4x+3$

and

$y_{2}=5x+44$ ?

Answer

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

To find the y-coordinate, substitute into one of the equations. Let's use $y_{1}$ :

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

Now, recall that the general form for a circle with center at $(x_{0},y_{0})$ is:

$(x-x_{0})^{2}+(y-y_{0})^{2}=r^{2}$

For our data, this means that our equation is:

$(x+41)^{2}+(y+161)^{2}=12^{2}$ or $(x+41)^{2}+(y+161)^{2}=144$

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Question

What is the radius of a circle with the equation ?

Answer

We need to expand this equation to $x^{2}-8x+y^{2}-6y=24$ and then complete the square.

This brings us to $x^{2}-8x+16+y^{2}-6y+9=24+16+9$ .

We simplify this to $(x-4)^{2}+(y-3)^{2}=49$ .

Thus the radius is 7.

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Question

A circle has its origin at . The point is on the edge of the circle. What is the radius of the circle?

Answer

The radius of the circle is equal to the hypotenuse of a right triangle with sides of lengths 5 and 7.

$r=\sqrt{74}$

This radical cannot be reduced further.

Compare your answer with the correct one above

Question

A circle with a radius of five is centered at the origin. A point on the circumference of the circle has an x-coordinate of two and a positive y-coordinate. What is the value of the y-coordinate?

Answer

Recall that the general form of the equation of a circle centered at the origin is:

$x^{2}+y^{2}=r^{2}$

We know that the radius of our circle is five. Therefore, we know that the equation for our circle is:

$x^{2}+y^{2}=5^{2}$

$x^{2}+y^{2}=25$

Now, the question asks for the positive y-coordinate when . To solve this, simply plugin for :

$2^{2}+y^{2}=25$

$4+y^{2}=25$

$y^{2}=21$

$y=\pm \sqrt{(21)}$

Since our answer will be positive, it must be $\sqrt{(21)}$ .

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Question

The following circle is moved $\frac{\pi}{2}$ spaces to the left. Where is its new center located?

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

Answer

Remember that the general equation for a circle with center and radius is .

With that in mind, our original center is at $(\frac{\pi}{2}, \frac{3\pi}{2})$ .

If we move the center $\frac{\pi}{2}$ units to the left, that means that we are subtracting $\frac{\pi}{2}$ from our given coordinates.

$\frac{\pi}{2} - \frac{\pi}{2} = 0$

$\frac{3\pi}{2} - \frac{\pi}{2} = \frac{2\pi}{2} = \pi$

Therefore, our new center is $(0, \pi)$ .

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Question

A square on the coordinate plane has vertices at the points with coordinates , , , and . Give the equation of the circle that circumscribes the square.

Answer

The equation of the circle on the coordinate plane with radius and center is

$(x-h)^{2} + (y-k)^{2} = r ^{2}$

The figure referenced is below:

The center of the circle is at the point of intersection of the diagonals, which, as is the case with any rectangle, bisect each other. Therefore, looking at the diagonal with endpoints and , we can set $x_{1} = y_{1} = 0 , x_{2} = y_{2} = 9$ in the midpoint formula:

$h =\frac{ x_{1}+ x_{2} }{2} = \frac{0+ 9}{2} = \frac{9}{2}$

and

$k=\frac{ y_{1}+ y_{2} }{2} = \frac{0+ 9}{2} = \frac{9}{2}$

The center of the circumscribing circle is therefore $\left ( \frac{9}{2}, \frac{9}{2} \right )$ .

The radius of the circumscribing circle is the distance from this point to any point on the circle. The distance formula can be used here:

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

Since we are actually trying to find $r ^{2}$ , we will use the form

$r^{2}= (x_{2} - x_{1})^{2} +(y_{2} - y_{1})^{2}$

Choosing the radius with endpoints and $\left ( \frac{9}{2}, \frac{9}{2} \right )$ , we set $x_{1} = y_{1} = 0 , x_{2} = y_{2} = \frac{9}{2}$ and substitute:

$r^{2}= \left ( \frac{9}{2} - 0 \right ) ^{2} + \left ( \frac{9}{2} - 0 \right ) ^{2}$

$= \left ( \frac{9}{2} \right )^{2} + \left ( \frac{9}{2} \right )^{2}$

$= \frac{81}{4 } + \frac{81}{4 }$

$= \frac{81}{2}$

Setting $h =k= \frac{9}{2}$ and $r^{2}= \frac{81}{2}$ and substituting in the circle equation:

$(x-h)^{2} + (y-k)^{2} = r ^{2}$

$\left ( x - \frac{9}{2}\right )^{2} + \left ( y- \frac{9}{2}\right )^{2} = \frac{81}{2}$ , the correct response.

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Question

The above figure shows a circle on the coordinate axes with its center at the origin. $\overarc {AB}$ has length $6 \pi$ .

Give the equation of the circle.

Answer

A $135^{\circ }$ arc of a circle represents $\frac{135}{360 } = \frac{3}{8}$ of the circle, so the length of the arc is three-eighths its circumference. Set up the equation and solve for :

$\frac{3}{8}C = 6 \pi$

$\frac{8} {3} \cdot \frac{3}{8}C =\frac{8} {3} \cdot 6 \pi$

$C = 16 \pi$

The equation of a circle on the coordinate plane is

$(x- h)^{2}+ (y - k)^2 = r^{2}$ ,

where are the coordinates of the center and is the radius.

The radius of a circle can be determined by dividing its circumference by $2 \pi$ , so

$r = \frac{C}{2 \pi } = \frac{16 \pi }{2 \pi } = 8$

The center of the circle is , so . Substituting 0, 0, and 8 for , , and , respectively, the equation of the circle becomes

$(x- 0)^{2}+ (y - 0)^2 =8^{2}$ ,

or

$x^{2}+ y^2 =64$ .

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Question

The above figure shows a circle on the coordinate axes with its center at the origin. The shaded region has area $49 \pi$ .

Give the equation of the circle.

Answer

The unshaded region is a $45 ^{\circ }$ sector of the circle, making the shaded region a $(360 - 45) ^{\circ } = 305 ^{\circ }$ sector, which represents $\frac{305}{360} = \frac{7}{8}$ of the circle. Therefore, if is the area of the circle, the area of the sector is $\frac{7}{8}A$ . The sector has area $49 \pi$ , so

$\frac{7}{8}A = 49 \pi$

Solve for :

$\frac{8} {7} \cdot \frac{7}{8}A =\frac{8} {7} \cdot 49 \pi$

$A =56\pi$

The equation of a circle on the coordinate plane is

$(x- h)^{2}+ (y - k)^2 = r^{2}$ ,

where are the coordinates of the center and is the radius.

The formula for the area of a circle, given its radius , is

$A = \pi r ^{2}$ .

Set $A = 56 \pi$ and solve for $r^{2}$ :

$\pi r ^{2} = 56 \pi$

$\frac{\pi r ^{2} }{\pi}=\frac{ 56 \pi}{\pi}$

$r ^{2} = 56$

The center of the circle is , so . Substituting 0, 0, and 56 for , , and $r^{2}$ , respectively, the equation of the circle becomes

$(x- 0)^{2}+ (y - 0)^2 =56$ ,

or

$x^{2}+ y^2 =56$ .

Compare your answer with the correct one above

Question

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?

Answer

The formula for the equation of a circle is:

$(x-h)^{2}+(y-k)^{2}=r^{2}$

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

and

and diameter = 6 therefore radius = 3

$(x-0)^{2}+(y-4)^{2}=3^{2}$

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

Compare your answer with the correct one above

Question

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?

Answer

The general equation of a circle is $(x-h)^{2}+(y-k)^{2}=r^{2}$ , 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 $\sqrt{29}$ .

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 $2\sqrt{29}$ .

Now, that we have circle A's new center and radius, we can write its general equation using $(x-h)^{2}+(y-k)^{2}=r^{2}$ .

$(x-(-2))^{2}+(y-2)^{2}=(2\sqrt{29})^{2}=2^{2}(\sqrt{29})^{2}=4(29)=116$ .

$(x+2)^{2}+(y-2)^{2}=116$ .

The answer is $(x+2)^{2}+(y-2)^{2}=116$ .

Compare your answer with the correct one above

Question

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

$y_{1}=4x+3$

and

$y_{2}=5x+44$ ?

Answer

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

To find the y-coordinate, substitute into one of the equations. Let's use $y_{1}$ :

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

Now, recall that the general form for a circle with center at $(x_{0},y_{0})$ is:

$(x-x_{0})^{2}+(y-y_{0})^{2}=r^{2}$

For our data, this means that our equation is:

$(x+41)^{2}+(y+161)^{2}=12^{2}$ or $(x+41)^{2}+(y+161)^{2}=144$

Compare your answer with the correct one above

Question

What is the radius of a circle with the equation ?

Answer

We need to expand this equation to $x^{2}-8x+y^{2}-6y=24$ and then complete the square.

This brings us to $x^{2}-8x+16+y^{2}-6y+9=24+16+9$ .

We simplify this to $(x-4)^{2}+(y-3)^{2}=49$ .

Thus the radius is 7.

Compare your answer with the correct one above

Question

A circle has its origin at . The point is on the edge of the circle. What is the radius of the circle?

Answer

The radius of the circle is equal to the hypotenuse of a right triangle with sides of lengths 5 and 7.

$r=\sqrt{74}$

This radical cannot be reduced further.

Compare your answer with the correct one above

Question

A circle with a radius of five is centered at the origin. A point on the circumference of the circle has an x-coordinate of two and a positive y-coordinate. What is the value of the y-coordinate?

Answer

Recall that the general form of the equation of a circle centered at the origin is:

$x^{2}+y^{2}=r^{2}$

We know that the radius of our circle is five. Therefore, we know that the equation for our circle is:

$x^{2}+y^{2}=5^{2}$

$x^{2}+y^{2}=25$

Now, the question asks for the positive y-coordinate when . To solve this, simply plugin for :

$2^{2}+y^{2}=25$

$4+y^{2}=25$

$y^{2}=21$

$y=\pm \sqrt{(21)}$

Since our answer will be positive, it must be $\sqrt{(21)}$ .

Compare your answer with the correct one above

Question

The following circle is moved $\frac{\pi}{2}$ spaces to the left. Where is its new center located?

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

Answer

Remember that the general equation for a circle with center and radius is .

With that in mind, our original center is at $(\frac{\pi}{2}, \frac{3\pi}{2})$ .

If we move the center $\frac{\pi}{2}$ units to the left, that means that we are subtracting $\frac{\pi}{2}$ from our given coordinates.

$\frac{\pi}{2} - \frac{\pi}{2} = 0$

$\frac{3\pi}{2} - \frac{\pi}{2} = \frac{2\pi}{2} = \pi$

Therefore, our new center is $(0, \pi)$ .

Compare your answer with the correct one above

Question

A square on the coordinate plane has vertices at the points with coordinates , , , and . Give the equation of the circle that circumscribes the square.

Answer

The equation of the circle on the coordinate plane with radius and center is

$(x-h)^{2} + (y-k)^{2} = r ^{2}$

The figure referenced is below:

The center of the circle is at the point of intersection of the diagonals, which, as is the case with any rectangle, bisect each other. Therefore, looking at the diagonal with endpoints and , we can set $x_{1} = y_{1} = 0 , x_{2} = y_{2} = 9$ in the midpoint formula:

$h =\frac{ x_{1}+ x_{2} }{2} = \frac{0+ 9}{2} = \frac{9}{2}$

and

$k=\frac{ y_{1}+ y_{2} }{2} = \frac{0+ 9}{2} = \frac{9}{2}$

The center of the circumscribing circle is therefore $\left ( \frac{9}{2}, \frac{9}{2} \right )$ .

The radius of the circumscribing circle is the distance from this point to any point on the circle. The distance formula can be used here:

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

Since we are actually trying to find $r ^{2}$ , we will use the form

$r^{2}= (x_{2} - x_{1})^{2} +(y_{2} - y_{1})^{2}$

Choosing the radius with endpoints and $\left ( \frac{9}{2}, \frac{9}{2} \right )$ , we set $x_{1} = y_{1} = 0 , x_{2} = y_{2} = \frac{9}{2}$ and substitute:

$r^{2}= \left ( \frac{9}{2} - 0 \right ) ^{2} + \left ( \frac{9}{2} - 0 \right ) ^{2}$

$= \left ( \frac{9}{2} \right )^{2} + \left ( \frac{9}{2} \right )^{2}$

$= \frac{81}{4 } + \frac{81}{4 }$

$= \frac{81}{2}$

Setting $h =k= \frac{9}{2}$ and $r^{2}= \frac{81}{2}$ and substituting in the circle equation:

$(x-h)^{2} + (y-k)^{2} = r ^{2}$

$\left ( x - \frac{9}{2}\right )^{2} + \left ( y- \frac{9}{2}\right )^{2} = \frac{81}{2}$ , the correct response.

Compare your answer with the correct one above

Question

The above figure shows a circle on the coordinate axes with its center at the origin. $\overarc {AB}$ has length $6 \pi$ .

Give the equation of the circle.

Answer

A $135^{\circ }$ arc of a circle represents $\frac{135}{360 } = \frac{3}{8}$ of the circle, so the length of the arc is three-eighths its circumference. Set up the equation and solve for :

$\frac{3}{8}C = 6 \pi$

$\frac{8} {3} \cdot \frac{3}{8}C =\frac{8} {3} \cdot 6 \pi$

$C = 16 \pi$

The equation of a circle on the coordinate plane is

$(x- h)^{2}+ (y - k)^2 = r^{2}$ ,

where are the coordinates of the center and is the radius.

The radius of a circle can be determined by dividing its circumference by $2 \pi$ , so

$r = \frac{C}{2 \pi } = \frac{16 \pi }{2 \pi } = 8$

The center of the circle is , so . Substituting 0, 0, and 8 for , , and , respectively, the equation of the circle becomes

$(x- 0)^{2}+ (y - 0)^2 =8^{2}$ ,

or

$x^{2}+ y^2 =64$ .

Compare your answer with the correct one above

Question

The above figure shows a circle on the coordinate axes with its center at the origin. The shaded region has area $49 \pi$ .

Give the equation of the circle.

Answer

The unshaded region is a $45 ^{\circ }$ sector of the circle, making the shaded region a $(360 - 45) ^{\circ } = 305 ^{\circ }$ sector, which represents $\frac{305}{360} = \frac{7}{8}$ of the circle. Therefore, if is the area of the circle, the area of the sector is $\frac{7}{8}A$ . The sector has area $49 \pi$ , so

$\frac{7}{8}A = 49 \pi$

Solve for :

$\frac{8} {7} \cdot \frac{7}{8}A =\frac{8} {7} \cdot 49 \pi$

$A =56\pi$

The equation of a circle on the coordinate plane is

$(x- h)^{2}+ (y - k)^2 = r^{2}$ ,

where are the coordinates of the center and is the radius.

The formula for the area of a circle, given its radius , is

$A = \pi r ^{2}$ .

Set $A = 56 \pi$ and solve for $r^{2}$ :

$\pi r ^{2} = 56 \pi$

$\frac{\pi r ^{2} }{\pi}=\frac{ 56 \pi}{\pi}$

$r ^{2} = 56$

The center of the circle is , so . Substituting 0, 0, and 56 for , , and $r^{2}$ , respectively, the equation of the circle becomes

$(x- 0)^{2}+ (y - 0)^2 =56$ ,

or

$x^{2}+ y^2 =56$ .

Compare your answer with the correct one above

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