Calculus: Analytic Geometry and Calculus, with Vectors

366 Cones and conics The formula (6.273) shows that the x and y axes are axes of symmetry of the hyperbola and that the origin i ...

6.2 Geometry of cones and conics 367 This equation is ponderous and nobody should ever dream of remembering it, but it is useful ...

368 Cones and con ics lies on the hyperbola, and we start by Y locating it in Figure 6.291. We then Ll M enter the wholesale ske ...

6.3 Ellipses 369 intersects the conic at two points, then the mid-point of the chord ofthe conic joining these points has coordi ...

(^370) Cones and conics focus and L2 is another directrix of the ellipse E. Thus E has two foci and two directrices. We can look ...

6.3 Ellipses 371 With the aid of Figure 6.31 and these formulas, we find that (6.334) IY1F,I + IF1F2I + IF2Y2I = 2IYrFLI + 177F- ...

372 and dividing by the right side gives the standard form (6.36) a2 + b2= 1, where b2 = a2(1 - e2) or (6.37) at = a2 - b2. Cone ...

6.3 Ellipses 373 2 The equation xz^2 22+3z=1 differs from equations of ellipses having their foci on the x axis because the deno ...

374 Cones and conics coordinate axes, find the equation of the line T1 tangent to the ellipse at P1. Ins.: XIX a2 + UZ = 1. 9 Fi ...

6.3 Ellipses 375 14 Figure 6.393 illustrates the fact that the line Y1P1 from a vertex Y1 to a point P1 on anellipse is parallel ...

376 Cones and conics 18 Study Figure 6.395 and discover the procedure by which the encircled points are determined, and then use ...

6.3 Ellipses 377 Show that (7) Show that (8) IvI = b2 + (a2 - b2) sin2 t. -X a --b2 sin t[a + X a2 - b2 cos t]. Letting )k be th ...

378 Cones and conics Hence (why?) P must lie on the line F1H2. Similarly (why?) P must lie on the line F22H2. This proves that t ...

6.4 Hyperbolas 379 to expressthis result in terms of coordinates, we can suddenly realize that it is almost familiar. If we put ...

380 Cones and conics A2 and A1. With the aid of Figure 6.41 and familiar facts about vectors tangent to spheres at their tips, w ...

6.4 Hyperbolas 381 which display information from the paragraph containing (6.281). Even though the equation of the hyperbola of ...

382 Cones and conics and for other quadrants we use symmetry. We can use the Pythagoras theorem to calculate the distance from t ...

6.4 Hyperbolas 383 4 The two vertices and one focus of a hyperbola are given. Describe a pro- cedure by which it is possible to ...

384 Cones and conics 10 Sketch a figure somewhat like Figure 6.491 and observe that L1 from a focus F perpendicular to the line ...

6.4 Hyperbolas 385 the hyperbola at Pl intersects the asymptotes of the hyperbola. fins.: The inter- sections with the asymptote ...