Chapter 6 Review of Conics The line D is called the directrix and the point F is called the focus If vertex is (h,k): (y-k)2 = 4a(x-h) Focus: (h+a,k) Directory: x=h-a (y-k)2 = -4a(x-h) Focus: (h-a,k) Directrix: x=h+a (x-h)2 = 4a(y-k) Focus: (k, h+a) Directrix: y = k-a (x-h)2 = -4a(y-k) Directrix: y = k+a A circle is a special ellipse with a=b. Major Axis Major Axis General Equation of a Conic: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 Parabola if B2 – 4AC = 0 Ellipse (or circle) if B2- 4AC < 0 Hyperbola if B2 - 4AC > 0

APPLICATIONS OF PARABOLAS If a light (or any other emitting source) is placed at the focus of a parabola, all the rays emanating from the light will reflect off the mirror in lines parallel to the axis of symmetry. Conversely, suppose rays of light (or other signals) emanate from a distant source (e.g. satellite) so that they are essentially parallel. When these rays strike the surface of a parabolic mirror whose axis of symmetry is parallel to these rays, they are reflected into a single point at the focus. Example 10 Satellite Dish If we model the satellite dish as an upward facing parabola, the formula is x2 = 4ay Satellite dish If we knew a point (x,y) on this parabola, we could solve for a. Use given information to find a point. Once we know a, we know the focus of a parabola in the form x2 = 4ay is (0,a). Therefore the receiver would need to be a feet away.

Applications of Ellipses Mean distance Aphelion Perihelion Major Axis Center The orbit of a planet about the Sun is an ellipse and the location of the Sun is the focus of the ellipse. p. 440 #77 The mean distance is 93 million miles so the length of the ellipse is 2 x 93 million = 186 million. The Aphelion is 94.5 million, so the Perihelion is 186 million – 94.5 million = 91.5 million miles. The equation for the ellipse is Recall that the vertices of a horizontal ellipse with center at the origin are on the major axis at the points (-a,0) and (a,0) where a is half the length of the ellipse (mean distance) and the foci are located at (-c,0) and (c,0). Therefore, in this problem, a = 93 million = 93 x 106 c = distance from center to sun = mean distance – perihelion = 93 million – 91.5 million = 1.5 million.

HOMEWORK p.424 #5,9,13 p. 485 #77 P. 488 #6 10 POINTS EXTRA CREDIT! p.487 Chapter Project #1