“A circle is the reflection of eternity. It has no beginning, and it has no end – and if you put several circles over each other, then you get a spiral. ”
– Maynard James Keenan.
How beautiful is it to see the circle as a reflection of eternity, right? However, it would be more wonderful if we could further understand and know the basics of a circle. This would enable us to see those perfect circles everywhere from a different perspective and appreciate how these things are made.
Enjoy this blog, and know the wonders of this endless but interesting conic section: CIRCLE.
What is a circle?
As defined by a blog from the internet, “A circle is a closed shape formed by tracing a point that moves in a plane such that its distance from a given point is constant. The word circle is derived from the Greek word kirkos, meaning hoop or ring. When a set of all points that are at a fixed distance from a fixed point are joined then the geometrical figure obtained is called a circle.”(Ashutosh, 2020)
The Circle as a Conic Section
The circle is the intersection of a plane perpendicular to the cone’s axis.
Moreover, the circle is called a conic section because they can be formed by intersecting a right circular cone.
The circle is the simplest and best-known conic section. The circle is the set of all points P(x,y) in a plane which moves a constant distance from a fixed point, known as it’s center. Additionally, the geometric definition of a circle is the path of all points that are at a constant distance r from the point (h, k) and form the circumference. Moreover, the distance r is the radius of the circle and the point C = (h, k) is the center of the circle. The diameter (D) is twice the length of the radius. (Introduction to Conic Sections, n.d.)
How do we find the radius and center of a circle from the standard equation ?
To define, the standard equation of the circle is (x-h)2+(y-k)2=r2 where in order to get the radius of the circle one must take the square root of r2. Additionally, to get the center of the circle, one must get the value of the variables h and k to form the ordered pair for the center of the circle.
Some Examples:
(x – 3)² + (y – 5) ² = 25
Center: (h , k) = (3 , 5)
Radius: √25 = 5 units
(x – 2)² + (y + 8) ² = 36
Center: (h , k) = (2 , -8)
Radius: √36 = 6 units
What is the general equation of a circle?
The general equation of the circle is Ax²+Cy²+Dx+Ey+F=0
An equation of the second degree in which the xy term is missing and the coefficients of the squared terms are always equal ( A=C).
How do we convert a circle’s Standard Equation to General Equation and vice versa?
Standard Equation to General Equation
Given the point (3,5) as the center of a circle and 5 units as its radius. We must first substitute these given values into the standard form formula then apply the proper steps to get its general equation.
For example
(x – 3)² + (y – 5) ² = 25
(x² – 6x + 9) + (y² – 10x + 25) = 25
x² – 6x + 9 + y² – 10y + 25 – 25 = 0
x² + y² – 6x – 10y +9 = 0
Write it in standard form.
Expand by squaring the group of x and y appropriately.
Transfer the constant to the other side
Arrange the equation. Then, this will be our final answer.
General Equation to Standard Equation
Given the general equation x² +y² -4x +16y +32 =0. Prior knowledge about completing the square is paramount in order to successfully convert this to a standard equation.
Completing the Square and Keeping the Balance
In order to complete the square, one must use the formula x² + bx + (b/2)². It is important to keep the balance of the equation and to do this, one should also add (b/2)² to the constant to keep both sides of the equation balanced out. To explain how (b/2)² came to be, refer to this picture for reference.
For Example
x² +y² -4x +16y +32 =0
(x² – 4x) + (y² + 16y) = -32
(x² – 4x + (4/ 2)² ) + (y² + 16y + (16/ 2)² ) = -32
(x² – 4x + 4 ) + (y² + 16y + 64 ) = -32 +4 +64
(x – 2)² + (y + 8) ² = 36
Begin with the General Equation. Ensure that the coefficients of the x² and y² values are equal.
Group the x and y values and transfer the constant to the other side.
Do complete the square
Keep the equation balanced
Factor out the equation and compute for the constant then this will be our final answer.
Graphing
Given (x – 3)² + (y – 5) ² = 25 . Graph the circle.
Center: (h , k) = (3 , 5)
Radius: √25 = 5 units
First, Plot the center of the graph then count 5 units to the left, right, above, and below. Then, connect these points in a circular way to form a circle.
Application
Circles can be found in everyday life, both in nature and in man-made things. The sun and moon are circle-shaped, the bases of mushrooms with domed tops are round. Many domestic goods, such as cups, candles, and doorknobs, are designed with circles. (“How Are Circles Used in Real Life?”). Circles appear frequently in architecture around the world. Architects often use the circle’s symmetrical properties when designing athletic tracks, recreational parks, buildings, roundabouts, Ferris-wheels, etc. Artists and painters find the circle almost indispensable in their work.
Circular cylinders are used to print newspapers. Technicians and Engineers take advantage of the circle’s symmetrical features, as seen by its application in watches, clocks, bicycles, cars, trains, ships, aircraft, radios, telephones, trolleys, wheelbarrows, air conditioning systems, and rockets, among other things. The features of circles are commonly used in building and housing construction. It is also used to create maps by taking measurements using rulers. (“Circles and Our Life”)
The most famous use of great circles in geography is for navigation because they represent the shortest distance between two points on a sphere. Due to the earth’s rotation, sailors and pilots using great circle routes must constantly adjust their route as the heading changes over long distances. (“Great Circles—Definition, and Examples in Geography”). To the Greeks, the circle was a symbol of divine symmetry, unity, and balance in nature. Greek mathematicians were fascinated by the geometry of circles and explored their properties for centuries. Because of technologies that utilize the shape circle, civilization has evolved significantly over the years.
The images show the Aeronautical Chart, GPS, Sea Chart, and Space Navigation and how they show circles.
Word Problem with Explanation and Solution
Cal wants to connect to the PisoNet Shop, but he wonders if the internet connection can reach their house. The WIFI satellite of the shop covers a 30-meters radius. The coordinates and radius of the satellite’s coverage is represented by the equation x² + y² -8x -12y -848. Cal’s house is located 20-meters east and 10-meters north of the satellite. What is the standard equation of the boundary of the internet service by the satellite? Does the internet reach Cal’s house?
Let’s Find Out!
Solution for the standard equation of the boundary of the internet service by the satellite:
Satellite’s coverage is represented by the equation x² + y² -8x -12y -848
x² + y² -8x -12y -848 = 0
(x² -8x) + ( y² -12y) = 848
(x² -8x +16) + ( y² -12y +36) = 848 +16 +36
(x -4)² + ( y -6)² = 900
Begin with the General Equation. Ensure that the coefficients of the x² and y² values are equal.
Group the x and y values and transfer the constant to the other side.
Do complete the square. Add 16 and 36 to both sides to keep the equation balanced.
Factor out the equation and compute for the constant then this will be our final answer.
Standard equation of the boundary of the internet service by the satellite:
(x -4)² + ( y -6)² = 900
Center/ coordinates of the satellite: (4,6)
Radius: 900 = 30 meters
Solution that proves that Cal’s house is within the satellites covered area:
Center/ coordinates of Cal’s house: (20,10)
Standard equation of the boundary of the internet service by the satellite: (x -4)² + ( y -6)² = 900
(x -4)² + ( y -6)² = 900
(20 -4)² + ( 10 -6)² ≟ 900
(16)² + ( 4)² ≟ 900
256 + 16 ≟ 900
272 < 900
Standard equation
Substitute the value of x and y by the coordinates of Cal’s house: (20,10).
Solve.
Change the equal sign into inequality sign <.
Since 272 is less than 900, this signifies that the squared distance from the satellite (center) to Cal’s house location is less than the squared radius of the internet service coverage area (272 meters vs. 900 meters) therefore we can conclude that the internet reaches Cal’s house because it is located inside the satellite’s covered area.
Another way that proves Cal’s house is within the satellites covered area:
This method uses the distance formula d = √(x₂-x₁)²+(y₂-y₁)² that involves:
Standard equation of the boundary of the internet service by the satellite: (x -4)² + ( y -6)² = 900
Radius: √900 = 30 meters
Center/ coordinates of the satellite: (4,6)
Center/ coordinates of Cal’s house: (20,10)
Whereas:
The first coordinates is the center of the satellite (4,6)
The second coordinates is the center of Cal’s house (20,10)
d = √(x2-x1)²+(y2-y1)²
d = √(20-4)²+(10-6)²
d = √(16)²+(4)²
d = √256+16
d = √272
d = 4√17
d ≈ 16.49 meters
Substitute the value of x₁ and y₁ by the first coordinates, the center of the satellite (4,6), and x₂ and y₂ by the second coordinates, the center of Cal’s house (20,10).
Solve.
Final answer.
Through the standard equation of the boundary of the internet service by the satellite (x -4)² + ( y -6)² = 900, we can say that the radius of the satellite’s covered area is √900 = 30 meters. Since the distance of Cal’s house from the satellite is 16.49 meters and is less than the radius of the satellite’s covered area which is 30 meters, therefore, we can conclude that Cal’s house is inside the covered area of the satellite thus the internet connection can reach Cal’s house.
Graph
Conclusion
In conclusion, a circle can be used to observe real-life situations and scenarios and it is important for our growth and development in society. For example, the concept or form of a circle can be utilized in the field of engineering and architecture as it can be used as a design aspect or base in making buildings. Another instance is the word problem above wherein Cal wonders if the internet can reach their house so in order to satisfy his curiosity he needs to solve it. Using the formula of a circle, he managed to solve and satisfy his curiosity that the internet can reach his house. In totality, understanding the concept of a circle is paramount as it has plenty of factors that we can use in various situations.
References
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Image Sources
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Pre-Calculus | XI-Owen Group IV 