Conic Sections Notes

Parabola

A parabola will only have either an x^2 term or a y^2 term, but not both. This means there is only one squared term in the equation.

A circle is defined by the coefficients of the x^2 and y^2 terms (A and C) being equal (A = C).
If A and C are both 10, 2, or -5 then it is a circle. The sign and value of A and C must be identical.
To analyze a circle, determine the center and the radius by completing the square.

An ellipse occurs when the coefficients A and C have the same sign (both positive or both negative) but are different numbers (A \neq C).
For example, A = 3 and C = 2, or A = 6 and C = 4.
To analyze, transform the equation to standard form after completing the square.
Key elements to identify: center, vertices, co-vertices, and foci.
The equation for finding the foci: c^2 = a^2 - b^2

A hyperbola is formed when the coefficients A and C have different signs (one positive, one negative).
To analyze, transform to standard form by completing the square.
Key elements to identify: center, vertices, and foci.
The equation for finding the foci: c^2 = a^2 + b^2

The general form of a conic equation is equal to zero.
A is the coefficient in front of the x^2 term and C is the coefficient in front of the y^2 term.
On an assessment, be prepared to identify the type of conic and provide a reason based on the coefficients A and C.

Equation: x^2 - 8x + y^2 + 6y + 9 = 0
A = 1 and C = 1 (both positive one), therefore, it is a circle because A = C.
To find the center and radius, complete the square:
- Group x terms and y terms: (x^2 - 8x + blank) + (y^2 + 6y + blank) = -9 + blank + blank
- Complete the square: (x^2 - 8x + 16) + (y^2 + 6y + 9) = -9 + 16 + 9
- Standard form: (x - 4)^2 + (y + 3)^2 = 16
The standard form of a circle is (x - h)^2 + (y - k)^2 = r^2

Equation: 5x^2 + 16y^2 - 250x - 192y - 399 = 0
A = 5 and C = 16; they are different positive numbers, so it's an ellipse.
The standard form of an ellipse is \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
Completing the square:
- Group terms: (5x^2 - 250x) + (16y^2 - 192y) = 399
- Factor out coefficients: 25(x^2 - 10x + blank) + 16(y^2 - 12y + blank) = 399 + blank + blank
- Complete the square: 25(x^2 - 10x + 25) + 16(y^2 - 12y + 36) = 399 + 2525 + 1636
To find the foci of the ellipse, use the formula: c^2 = a^2 - b^2

Equation: y^2 + 2x - x^2 - 4y = -7
Rearrange to general form: -x^2 + y^2 + 2x - 4y + 7 = 0
A = -1 and C = 1; they have different signs, so it's a hyperbola.
General form: \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
Completing the square:
- Group terms: (x^2 - 2x + blank) - (y^2 + 4y + blank) = -7 + blank
- Factor out -1: (x^2 - 2x + blank) -1(y^2 + 4y + blank) = -7 + blank
To find the foci of the hyperbola, use the formula: c^2 = a^2 + b^2

Equation: x^2 - 2x = -y - 9
Only an x^2 term exists; therefore, it is a parabola.
Rearrange to standard form by completing the square:
- (x^2 - 2x + blank) = -y - 9 + blank
- The standard form of a parabola with a vertical axis is (x - h)^2 = 4p(y - k)
Vertex: (h, k)
If x is squared, the parabola opens up or down.
If 4p is positive, opens up; if negative, opens down.
Find p by setting 4p equal to the coefficient of (y - k).
Focus is p units inside the curve and the directrix is a line p units outside the curve.
Lattice rectum: The length across the focus, which helps determine the width of the parabola, is equal to 4p