. Plane and solid analytic geometry . as center and AQ a,s . , radius describe a circle, and ^ ^ ■ with F as center and AQ as Fig. 3 i • i radius describe a second circle. The points of intersection of these two circles will lie on the ellipse, since the sum of the radii is AQ + AQ=2a. It is, of course, not necessary to draw the complete circles,but only so much of them as to determine their points of in-tersection. Moreover, four points, instead of tivo, can be ob-tained from each pair of settings of the compasses by simplyreversing the roles of F and F\ EXERCISES X 1- Construct the ellipse f
. Plane and solid analytic geometry . as center and AQ a,s . , radius describe a circle, and ^ ^ ■ with F as center and AQ as Fig. 3 i • i radius describe a second circle. The points of intersection of these two circles will lie on the ellipse, since the sum of the radii is AQ + AQ=2a. It is, of course, not necessary to draw the complete circles,but only so much of them as to determine their points of in-tersection. Moreover, four points, instead of tivo, can be ob-tained from each pair of settings of the compasses by simplyreversing the roles of F and F\ EXERCISES X 1- Construct the ellipse for which c — 2\ cm., a = 4 cm. 2. Erom the ellipse just constructed make a templet, withholes at the foci and with the axes properly drawn. ^. 3. Construct the ellipse whose axes are 4 cm. and 6 cm. 3. Equation of the Ellipse. It is natural to choose the axesof the ellipse as the coordinate axes (Eig. 4). Let the foci lie THE ELLIPSE 105 on the axis of x, and let F: {x, y) be any point of the , from (1), § 1, . y _. Fig. 4 (1) V(a;-c)2+2/2 + V(x + c)2+ y^ = 2a. Transpose one of the radicals and square: (x — cy-\- 2/2= (a; + 0)2+ 2/^ — 4a^{x -j- c)2 + 2/2 + 4a. Hence (2) a V(aj + c)2 + 2/2 = a^ -f ca;. To remove this radical, square again : (3) a2a;2 + 2 aca; + a2c2 ^ ^2^2 = ^4 _|_ 2 ^2^0; -f c2a;*^,or (a2 — c2)a;2 -\- a^y^ = a2(a2 — c2). But, by (2), § 1, ai-c = h\ and hence (4) 6^x2 + aY = a^ft*, or (5) ^ + ■=1. ^2 a^ This is the standard form of the equation of the ellipse, re-ferred to its axes as the axes of coordinates. The proof, how-ever, is not as yet complete, for it remains to show, conversely,that any point {x, y) whose coordinates satisfy equation (5)is a point of the ellipse. To do this, it is suflScient to showthat x,y satisfy (1). From (5) we mount up to (4) aud thenceto (3), since all of these are equivalent equations. When, 106 ANALYTIC GEOMETRY however, we extract a square root we obtain two equationseach time, and so we are l
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