8.2: CIRCLES AND ELLIPSES
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1 8.: CIRCLES AND ELLIPSES GEOMETRY OF AN ELLIPSE Geometry of n Ellipse Definition: An ellipse is the set of ll points in plne whose distnce from two fixed points in the plne hve constnt sum. Voculry The fixed points re the foci of the ellipse. The line through the foci is the focl xis. The point on the focl xis midwy etween the foci is the center. The points where the ellipse intersects its xis re the vertices. d 1 d F 1 1 F d d is constnt 1
2 GEOMETRY OF AN ELLIPSE More voculry A line segment with endpoints on n ellipse is chord. The chord lying on the focl xis is the mjor xis (the length is ). The chord through the center perpendiculr to the focl xis is the minor xis (the length is ). The numer is the semimjor xis, nd the numer is the semiminor xis. PICTURES BY DEFINITION Focus (x, y) P(x, y) Focus d 1 d (F 1, 0) (F, 0)
3 PICTURES -EXPANDED Center Focus Vertex (0, ) Minor Axis Focus Mjor Axis (-, 0) (-c, 0) (0, 0) (c, 0) (, 0) (0, -) Vertex is the SEMI-MAJOR xis is the SEMI-MINOR xis ELLIPSES - CENTER AT (0,0) St. Fm. Focl xis Foci Semi-Mjor Semi-Minor Pyth. Rel. x x xis y 1 c,0 c y xis y 1 0, c x c 3
4 ELLIPSES - CENTER AT (H, K) St. fm. Focl xis Foci Semi-Mjor Semi-Minor Pyth. Rel. xh yk 1 y k h c, k c yk xh 1 x h hk, c c Exmple 1: Grph the following ellipse x y ( hk, ) (,) 4, c F c c 4 foci :, -5-5 C F - - 4
5 Exmple : Determine the eqution of the ellipse with foci t (3,5) nd (9,5) nd minor xis of length 10. c is hlf the distnce etween foci, so c=3 The center is the midpoint etween the foci, so (6,5) 10 5 c x6 y F 1 F Exmple 3: Sketch the grph of the ellipse with the eqution elow, then determine the foci, vertices, semimjor xis, nd semiminor xis. 4x 5y x y 5 4 x y x 5y 100 c 5 4 c 1 c -5F F 5 1 foci: 1,0 vertices: 5,0 semimjor xis:5 semiminor xis: - 5
6 Exmple 4: Sketch the grph of the ellipse with the eqution elow, then find the foci, vertices, semimjor xis, nd semiminor xis. 5x 10x4y 8y5 0 x x y y x x y y x1 y1 x y x y x1 y Semimjor xis = 7 14,verticl Semiminor xis = ,horizontl c 7 14 c 5 70 c 10 foci: 1, F 1 F - 6
7 ECCENTRICITY AND ORBITS Definition: suppose tht n ellipse hs semimjor xis nd focl length c. Then the eccentricity e of the ellipse is defined s e c Note tht since the focl length for n ellipse must e less thn the semimjor xis, the eccentricity must e etween 0 nd 1. (or more specificlly 0,1) Exmple 5: Determine the eccentricity of the ellipse in exmple , c c 5 e Exmple 6: Show tht n ellipse of eccentricity 0 is circle If e=0, then c=0, therefore 0 Thus, the semimjor nd semiminor xes hve the sme length, nd the ellipse is circle. 7
8 Exmple 7: According to Kepler s first lw, every plnet orits the sun in n ellipticl orit, with the sun t one focus. The eccentricity of the erth s orit out the sun is pproximtely The closest distnce etween the erth nd sun is pproximtely 93 million miles. Wht is the furthest distnce etween the erth nd the sun? Let e the semimjor xis of the orit. Assume tht the center of the orit is (0,0), nd the sun is t the focus (c,0). c93 c c e c 93 c0.0167c c c Sun -c distnce: cc million miles ELLIPSOIDS OF REVOLUTION Rotte ellipse out its focl xis to get n ellipsoid of revolution. Exmples of these include whispering glleries nd lithotripter, device which uses shockwves to destroy kidney stones. 8
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