conic section - ellipse

• An ellipse is formed by a plane intersecting a cone at an angle to its base.
• An ellipse is a set of all points P on a plane such that the sum of its distances from two fixed points is a constant.
• The two fixed points, usually denoted by 𝑓 1 and 𝑓 2 , are called the foci.
• The center of an ellipse is the point in the middle of an ellipse and is denoted by (��, 𝑘).
• The foci of an ellipse are two points located on the major axis and have the same distance from the center.
• The horizontal distance from center to graph of an ellipse is denoted by 𝑎.
• The vertical distance from center to graph of an ellipse is denoted by 𝑏.
• The major axis of an ellipse is the longest segment across the graph of an ellipse with endpoints called the vertices.
• The minor axis of an ellipse is a segment perpendicular to and bisecting the major axis, with endpoints on the ellipse.
• A horizontal ellipse is an ellipse wherein the major axis is horizontal.
• If 𝑎 > 𝑏, then the length of the major axis is |2𝑎|, and the length of the minor axis is |2𝑏|.
• The vertices of a horizontal ellipse are (ℎ + 𝑎, 𝑘) and (ℎ − 𝑎, 𝑘).
• The length of the major axis of an ellipse is represented as 𝑎𝑛𝑑.
• The vertices of an ellipse are represented as (𝑎, 0) and (−𝑎, 0).
• According to the definition of an ellipse, the sum of the distances from the foci to any point on the ellipse must be constant, represented as �� = √ (𝒙 − ��) 𝟐 + 𝒚 𝟐 + √ (�� + ��) 𝟐 + 𝒚 𝟐.
• The distance from the center of an ellipse to a point on the graph is represented as �� for the horizontal distance and 𝑏 for the vertical distance.
• The endpoints of the minor axis of an ellipse are represented as (0, −��) and (0, 𝑏).
• The equation of an ellipse in terms of �� and 𝑏 can be found by combining the distance formula with the definition of an ellipse.
• The distance between a point on an ellipse and a focus can be calculated using the distance formula.
• The distance between a point on an ellipse and the other focus can also be calculated using the distance formula.
• The foci of an ellipse are represented as (−��, 0) and (𝑐, 0).
• The coordinates of the foci of an ellipse are (−𝑐, 0) and (𝑐, 0).
• Examples of graphing an ellipse include graphing the equations: (𝑥 − ��)2 𝑎2 + (𝑦 − 𝑘)2 ��2 = 1, (�� − 𝑎)2 𝑎2 + (𝑦 − 𝑘)2 𝑏2 = 1, and (𝑥 − 𝑎)2 𝑎2 + (𝑦 − 𝑘)2 𝑏2 = 1.
• The endpoints of the minor axis of a horizontal ellipse are (ℎ, �� + 𝑏) and (ℎ, 𝑘 − 𝑏).
• To graph an ellipse, transform the given equation to standard form, determine ℎ, 𝑘, 𝑎, and 𝑏, locate the center (ℎ, 𝑘), locate the endpoints of the major axis and the minor axis, and draw the ellipse that passes on these four points and label the graphs (center, major axis, minor axis, and endpoints of the axes).
• If 𝑎 < 𝑏, then the length of the major axis is |2𝑏|, and the length of the minor axis is |2𝑎|.
• The endpoints of the minor axis are (𝑎 + 𝑎, 𝑘), (𝑎 − 𝑎, 𝑘).
• The vertices of a vertical ellipse are (��, 𝑘 + 𝑏), (��, �� − 𝑏).
• The standard equation of an ellipse with center at (h, k) is: (𝑥 − ℎ)2 ��2 + (𝑦 − 𝑘)2 ��2 = 1.
• A vertical ellipse is an ellipse wherein the major axis is vertical.
• If 𝑎 < 𝑏, then the length of the major axis is |2𝑎|, and the length of the minor axis is |2𝑏|.
• The vertices of a vertical ellipse are (ℎ + ��, 𝑘) and (ℎ − 𝑎, 𝑘).
• The endpoints of the minor axis of a vertical ellipse are (ℎ, �� + 𝑏) and (ℎ, 𝑘 − 𝑏).
• The vertices of the ellipse are (ℎ, ��) = (𝟕, −𝟏) and (ℎ − ��, 𝑘) = (−𝟑, −𝟏).
• The equation for the ellipse in standard form is: 𝒙𝟐𝟐𝟐 + 𝒚����𝟐 = 𝟏.
• The length of the major axis is |2��| = 𝟏𝟎.
• The horizontal distance from the center to the graph is 5.
• The center of the ellipse in standard form is located at (ℎ, ��) = (𝟎, 𝟎).
• The vertices of the ellipse in standard form are (ℎ, �� + 𝑏) = (��, 𝟓) and (ℎ, �� − 𝑏) = (𝟎, −𝟓).
• The vertical distance from the center to the graph is 3.