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.

Circle

• 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.

Ellipse

• 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: $c^2 = a^2 - b^2$

Hyperbola

• 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: $c^2 = a^2 + b^2$

General Form and Identifying Conics

• 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.

Example 1: Identifying a Circle

• 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 $(x - h)^2 + (y - k)^2 = r^2$

Example 2: Identifying an Ellipse

• 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: $25(x^2 - 10x + 25) + 16(y^2 - 12y + 36) = 399 + 25<em>25 + 16</em>36$

• To find the foci of the ellipse, use the formula: $c^2 = a^2 - b^2$

Example 3: Identifying a Hyperbola

• 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: $c^2 = a^2 + b^2$

Example 4: Identifying a Parabola

• 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 $4p$