If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

# Welcome to Conic Sections

I greet you this day,
Second: view the videos.
Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained examples.
Fifth: check your solutions with the calculators as applicable.
I wrote the codes for some of the calculators using JavaScript, a client-side scripting language. Please use the latest versions of Internet browsers. The calculators should work.
The Wolfram Alpha widgets (many thanks to the developers) were used for some calculators.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting.

Samuel Dominic Chukwuemeka (SamDom For Peace) B.Eng., A.A.T, M.Ed., M.S

## Circle

### Circle

A Circle is defined as the locus of points equidistant (equal distance) from a fixed point.
The fixed point is the center of the circle.
The equal distance is the radius of the circle.

The equation of a circle can be written in any of these two forms:
Standard Form
$(x - h)^2 + (y - k)^2 = r^2$
where:
$x, y$ are the variables
$(h, k)$ are the coordinates of the center of the circle
$r$ is the radius of the circle

General Form
$x^2 + y^2 + cx + dy + e = 0$
where:
$x, y$ are the variables
$c$ is the coefficient of $x$
$d$ is the coefficient of $y$
$c, d, e$ are values/constants

NOTE For:
(1.) $(x - h)^2 + (y - k)^2 = r^2$; Center = $(h, k)$, Radius = $r$

(2.) $(x - h)^2 + (y + k)^2 = r^2$; Center = $(h, -k)$, Radius = $r$

(3.) $(x + h)^2 + (y - k)^2 = r^2$; Center = $(-h, k)$, Radius = $r$

(4.) $(x + h)^2 + (y + k)^2 = r^2$; Center = $(-h, -k)$, Radius = $r$

(5.) $(x - h)^2 + (y - k)^2 = e$; Center = $(h, k)$, Radius = $\sqrt{e}$

(6.) $(x - h)^2 + (y + k)^2 = e$; Center = $(h, -k)$, Radius = $\sqrt{e}$

(7.) $(x + h)^2 + (y - k)^2 = e$; Center = $(-h, k)$, Radius = $\sqrt{e}$

(8.) $(x + h)^2 + (y + k)^2 = e$; Center = $(-h, -k)$, Radius = $\sqrt{e}$

(9.) When converting from Standard Form to General Form, expand. Multiply the binomials and arrange the terms in order.

(10.) When converting from General Form to Standard Form, Completing the Square method is used. Then, arrange the terms in order.

## Calculators

### Calculators for Circles

All input values should be integers or decimals only. No fractions
Some of the output values are in fractions and/or radicals. Pick the one you need.
Simplify further as necessary.
• Given: Standard Form of the Equation of a Circle
To Find: other details

$(x - $$)^2 \:\:+\:\: (y -$$)^2$ $\:\:=\:\:$ $^2$

The center is (, )

The General Form is:
$x^2$ $+$ $y^2$ $+$ $x$ $+$ $y$ $+$ $= 0$

• Given: General Form of the Equation of a Circle
To Find: other details

$x^2$ $+$ $y^2$ $+$ $x$ $+$ $y$ $+$ $= 0$

The center is (, ) OR (, )

The Standard Form is:
$(x - $$)^2 \:\:+\:\: (y -$$)^2$ $\:\:=\:\:$ $^2$

• Given: Center, Radius of a Circle
To Find: other details

The center is (, )

The Standard Form is:
$(x - $$)^2 \:\:+\:\: (y -$$)^2$ $\:\:=\:\:$ $^2$

The General Form is:
$x^2$ $+$ $y^2$ $+$ $x$ $+$ $y$ $+$ $= 0$

• Given: Center, Point on the Circumference of a Circle
To Find: other details

The center is (, )

1st endpoint of diameter is (, )

The diameter is OR

2nd endpoint of diameter is (, )

The Standard Form is:
$(x - $$)^2 \:\:+\:\: (y -$$)^2$ $\:\:=\:\:$ $^2$

The General Form is:
$x^2$ $+$ $y^2$ $+$ $x$ $+$ $y$ $+$ $= 0$

• Given: End Points of the Diameter of a Circle
To Find: other details

1st endpoint of diameter is (, )

2nd endpoint of diameter is (, )

The center is (, ) OR (, )

The diameter is OR

The Standard Form is:
$(x - $$)^2 \:\:+\:\: (y -$$)^2$ $\:\:=\:\:$ $^2$

The General Form is:
$x^2$ $+$ $y^2$ $+$ $x$ $+$ $y$ $+$ $= 0$

• Given: Any Two Points
To Find: Point on the $y-axis$ Equidistant from the Two Points

1st point is (, )

2nd point is (, )

The point on the $y-axis$ equidistant from the two points is (, )

• Given: Any Two Points
To Find: Point on the $x-axis$ Equidistant from the Two Points

1st point is (, )

2nd point is (, )

The point on the $x-axis$ equidistant from the two points is (, )

### References

Chukwuemeka, S.D (2019, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.chukwuemekasamuel.com

Bittinger, M. L., Beecher, J. A., Ellenbogen, D. J., & Penna, J. A. (2017). Algebra and Trigonometry: Graphs and Models ($6^{th}$ ed.). Boston: Pearson.

Coburn, J., & Coffelt, J. (2014). College Algebra Essentials ($3^{rd}$ ed.). New York: McGraw-Hill

Sullivan, M., & Sullivan, M. (2017). Algebra & Trigonometry ($7^{th}$ ed.). Boston: Pearson.

Authority (NZQA), (n.d.). Mathematics and Statistics subject resources. www.nzqa.govt.nz. Retrieved December 14, 2020, from https://www.nzqa.govt.nz/ncea/subjects/mathematics/levels/

CMAT Question Papers CMAT Previous Year Question Bank - Careerindia. (n.d.). Retrieved May 30, 2020, from https://www.careerindia.com/entrance-exam/cmat-question-papers-e23.html