Isaiah's Math Lessons - Circles in Coordinate Geometry

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Circle Definition:The locus of a point that moves such that its distance from a fixed point called the center is constant. The constant distance is called the radius, r of the circle.
General Equation:
Ax² + Ay² + Dx + Ey + F = 0
Or
x² + y² + Dx + Ey + F = 0

Standard Equation:
Circle with center at (h, k) : (x−h)² + (y−k)² = r²
Circle with center at (0, 0) : x² + y² = r²

NOTE: The r variable here stands for the radius of the circle.

Examples and Detailed Solutions:

1. What is the general equation of a circle with center at (1, -2) and radius equal to 3?

Based from the Standard equation of the Circle, we will have the below equation:
(x−h)² + (y−k)² = r²
But (h, k) is (1,-2) and r = 3
(x−1)² + (y−(−2))² = 3²
(x−1)² + (y+2)² = 3²
We know that (a+b)² = a² + 2ab + b²
x² − 2x + 1 + y²+ 4y + 4 = 9
Rearranging the Equation, we will have x² + y² - 2x + 4y – 4 = 0

2. Find the equation in standard from, center and radius of a circle with equation 4x²+ 4y²−4x+ 12y−15=0

The trick here is to first make the coefficient of x² and y² be equal to 1.
Therefore, we divide the whole equation by 4. We divide it by 4 since that is the coefficient of both x² and y².
(4x²+ 4y²−4x+ 12y−15) / 4 = 0 = > x²+ y²− x+3y− 15/4 = 0

After making the coefficient of x2 and y2 equal to 1, we now group similar terms then completing the square
( x2 − x + _ ) + ( y2+ 3y+ _ ) = 15/4 + _ + _
How do we complete a square? First is we divide the coefficient of the middle term by 2 then square it. For the left side of the equation, the coefficient of the middle term is -1 which is the coefficient of x. We then divide -1 by 2 and then square it which results to 1/4. Same thing goes for the y terms. We divided 3 by 2 then square it which will result to 9/4.

( x²− x + 1/4) + ( y² + 3y + 9/4) = 15/4 + 1/4 + 9/4
We then factor it to a binomial raised to the 2nd power. We copy the middle term of the expressions above.
(x−1/2)² + (y+3/2)² = ( 5/2)²

Center (1/2, -3/2) and r = 5/2

Additional Shortcut: How to find the Center of a Circle in General form having the equation
x² + y² + Dx + Ey + F = 0
Shortcut Formula (Without Completing the Square)
Center (−D/2 , -E/2)
Radius = 1/2√(D²+ E² − 4F)

Practice Problems:
1. What is the center of the circle with equation x²+ y²+ 4x−6y+9=0?
2. For what value of K will make x²+ y²+ 4x−6y+K=0 a point circle? (Clue : a point circle is a circle with radius = 0 )
3. Find the equation of the circle tangent to the x-axis with center at (1, 2)?
4. What is the value of the eccentricity of a circle? What does eccentricity means?

Answers:
1. C (-2,3)
2. K = 13
3. (x−1)² + (y−2)² = 2²
4. e=0, it is defined as the parameter which is associated with every conic section. It can be thought as a measure of how much the conic deviates to become a circular figure. Other values of eccentricity are Parabola (e=1), Ellipse (0 < e < 1) and Hyperbola (e > 1).