Chapter 10 Conic Sections 10.1 – The Parabola and the Circle 10.2 – The Ellipse 10.3 – The Hyperbola 10.4 – Nonlinear Systems of Equations and Their Applications

Recall: Distance & Midpoint Formulas Circles & Parabolas 10.2 The Ellipse

Graph Ellipses An ellipse is a set of point in a plane, the sum of whose distances from two fixed points is a constant. The two fixed points are called the foci (each is a focus) of the ellipse. F1 and F2 represent the two foci of an ellipse The standard form of an ellipse with its center at the origin: x-intercepts : (a, 0) & (-a, 0) y-intercepts: (0, b) and (0, -b) If a2 > b2, the major axis is along the x-axis. If b2 > a2, the major axis is along the y-axis.

Graph Ellipses Example 1: Graph the ellipse and label the x- and y-intercepts. (–4, 0) (4, 0) (0, –2) (0, 2) a = 4, x-intercepts (Vertices): (4, 0) & (-4, 0). b = 2, y-intercepts: (0, 2) & (0, -2) Find the area of the ellipse. 𝐴=𝜋𝑎𝑏 𝐴=𝜋 4 2 =8 𝜋

Graph Ellipses Example 2: a = 3, x-intercepts (Vertices): (3, 0) & (-3, 0). b = 2, y-intercepts: (0, 2) & (0, -2) Find the area of the ellipse. 𝐴=𝜋𝑎𝑏 𝐴=𝜋 3 2 =6 𝜋

Graph Ellipses Example 3: Solution: To Write the equation in standard form, divide both sides by 1764. (0, 7) (6, 0) (0, -7) (-6, 0) a = 6 b = 7 The x-intercepts are (6, 0) and (6, 0). The y-intercepts(Vertices) are (0, 7) and (0, 7).

Graph Ellipses with Centers at (h, k) In the formula, h shifts the graph left or right from the origin and k shifts the graph up or down from the origin. Example 4: Graph the ellipse. Label the center and the points above and below, to the left and to the right of the center. C(h, k) = C(2, 1) a = 4, b = 5 4 units to the right and left of (2, 1): (6, 1) & (2, 1). 5 units above and below (2, 1): Vertices: (2, 6) & (2, 4)

Example 5: Graph Ellipses with Centers at (h, k) Label the center and the points above and below, to the left and to the right of the center. C(2, -3) a = 5 and b = 4 Vertices: (3, 3) & (7, 3) (2, 7 ) & (2, 1) Find the area of the ellipse. 𝐴=𝜋𝑎𝑏 𝐴=𝜋 5 4 =20 𝜋

Example 6: A communication satellite travels in an elliptical orbit around Earth. The maximum distance from earth is 25,000 miles and the minimum distance is 24,500 miles. Earth is at one focus of the ellipse. Find the distance from earth to the other focus. 𝑆𝑎𝑡𝑒𝑙𝑙𝑖𝑡𝑒 𝐹 1 𝐴 2 =25,000 𝑚𝑖𝑙𝑒𝑠 𝐹 1 𝐴 1 =24,500 𝑚𝑖𝑙𝑒𝑠 𝐴 1 −−− − 𝐹 1 −−−−−−−−−−− 𝐹 2 -------- 𝐴 2 Earth 𝐹 1 𝐹 2 =25,000 −24,500=500 𝑚𝑖𝑙𝑒𝑠

Summary: Distance & Midpoint Formulas Circles Parabolas Ellipses Area of the ellipse: 𝐴=𝜋𝑎𝑏