Graphing circles | Geometry | Quantitative reasoning

Graphing circles

Just as there is an equation for a straight line, there is also an equation for a circle. However, the equation for a circle is a bit more complicated than $y = m x + b$ . It can be described as $(x - a)^{2} + (y - b)^{2} = r^{2}$ where $a$ and $b$ represent the $x$ and $y$ coordinates of the center point of the circle, respectively, and $r$ represents the radius of the circle.

Please note that the $x$ and $y$ values should not be replaced with the coordinates of the center point. These coordinates should exclusively replace the $a$ and $b$ variables. The pairs of $x$ and $y$ values represent the coordinates of the infinite points along the circle, not the center of the circle.

Interestingly enough, if the center of the circle is located at the origin, the equation simplifies significantly. A circle with a radius of $2$ with a center point exactly where the $x$ and $y$ axes meet would be $x^{2} + y^{2} = 2$ . This is because $a$ and $b$ each amount to zero in the equation because the coordinates of the center point are $(0, 0)$ . Try using the equation for a circle to solve this problem.

$(x - a)^{2} + (y - b)^{2} = r^{2}$

Which of the following equations represents the circle drawn below? The two points represent the highest and lowest points of the circle.

Answers:
A. $(x + 5)^{2} + (y - 4)^{2} = 36$
B. $(x + 5)^{2} + (y - 4)^{2} = 3$
C. $(x - 5)^{2} + (y + 4)^{2} = 9$
D. $(x + 5)^{2} + (y - 4)^{2} = 9$
E. $(x - 5)^{2} + (y + 4)^{2} = 36$

(spoiler)

Correct Answer: D. $(x + 5)^{2} + (y - 4)^{2} = 9$

Start by finding the coordinates of the center of the circle. Even though the center is not directly indicated, you can find the center by finding the middle points between the top and bottom of the circle. The point between these two points must be $(- 5, 4)$ because these are the averages of the $x$ values and the $y$ values, respectively.

Next, you just need to find the radius of the circle. The radius would be the distance between the center and any of the two points. This distance would be $3$ because the $y$ coordinate of the central point, $4$ , is $3$ away from both the $y$ values of the top and bottom points, $7$ and $1$ .

With a central point located at $(- 5, 4)$ and a radius of $3$ , you can simply plug this information into the equation for a circle. Remember, a and b represent the $x$ and $y$ coordinates of the central point.

$(x - (- 5))^{2} + (y - 4)^{2} = r^{2}$

$(x + 5)^{2} + (y - 4)^{2} = 3^{2}$

$(x + 5)^{2} + (y - 4)^{2} = 9$