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Criswell PreCalculus
Conics (Parabola, Circle, Ellipse, Hyperbola) – Chapter 3
| Question | Answer |
|---|---|
| Midpoint | MP = ( ave x values , ave y values ) MP = ( [x1+x2] / 2 , [y1+y2] / 2 ) |
| Distance Formula | d= √ [ (y2–y1)^2 + (x2–x1)^2 ] d=√ [ (∆y )^2 + (∆x)^2 ] |
| Locus of a parabola | The essence of a parabola is found with the relationship between the focus and the directrix. The collection of points that are equal distance from the focus to (x , y) and then perpendicular to the directrix is what creates the parabolic shape. |
| Equation of Parabola in "Vertex Form" or (h , k) form | y = a (x –h)^2 +k "vertical" or x = a (y –k)^2 +h "horizontal" |
| Focus of parabola | The Focus is always located in the interior of a parabola and is p units away from the vertex. a = 1 / (4p) |
| Directrix of a parabola | The Directrix is always located on the exterior of a parabola and is p units away from the vertex. |
| Locus of a Circle | The essence of a circles is found in the collection of points on a curve that are equidistance from a set point. This set point is the center of the circle. |
| Equation of a circle in (h , k) form | (x-h)^2 + (y-k)^2 = r^2 |
| Locus of an ellipse | The collection of all points on a curve that generate an equal distance from two set points located on the major axis known as foci. This distance is equal to the major axis length 2a. Any (x , y) on the curve will satisfy the condition above. |
| Equation of an ellipse in (h , k) form | [(x-h)^2 ] / a^2 + [(y-k)^2] / b^2 =1 Whichever denominator is larger determines the orientation of the ellipse as "horizontal" or "vertical" |
| Equation to be used when calculating the placement of foci from the center point. | a^2 - b^2= c^2 or Larger denominator - Smaller denominator = c^2 |
| Finding the center of an ellipse | 1) The mid-point of the major axis endpoints "vertices". 2) The mid-point of the minor axis endpoints "co-vertices". 3) The mid-point of the foci coordinates. |
| Area of a Circle | A = π*r^2 |
| Circumference or perimeter of a Circle | C = 2π*r |
| Area of a Ellipse | A = π * a* b |
| Circumference or perimeter of an Ellipse | C = 2π *√ ( (a^2+b^2) / 2 ) |
| The Eccentricity of ellipse | e = c/a e should fall between 0 and 1 if e = 0, then the ellipse is actually a circle. |
| Equation of an Hyperbola in (h , k) form | [(x-h)^2 ] / a^2 – [(y-k)^2] / b^2 =1 [(y-k)^2 ] / a^2 – [(x-h)^2] / b^2 =1 Whichever term is listed first determines orientation of hyperbola |
| Eccentricity of Conic Sections | e = c/a if e = 0, then the conic a circle. if e = 1, then the conic a parabola if e< 1, then the conic an ellipse. if e >1, then the conic a hyperbola. |
| Locus of an Hyperbola | The collection of all points on a curve that generate distances from points located on the transverse axis known as foci where the difference in distances is equal to the transverse length 2a. Any (x , y) on the curve will satisfy the condition above. |
Created by:
Troy.Criswell
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