Calculus: Analytic Geometry and Calculus, with Vectors

385 Cones and conic, Aristotle told Alexander the Great that there is no royal road to geometry, but it is quite easy to see wha ...

6.4 Hyperbolas 387 then P traverses a branch of a hyperbola. Calculate the acceleration a and show that a = k2r. 22 Copy Figure ...

388 Cones and conics 25 Let 0 < b < a. Prove that when X < b2, the ellipse having the equation 2 y2 a2-X b2-X has its f ...

6.4 Hyperbolas 389 30 Let the angle A10A2 of Figure 6.497 be a given angle between 0 and a. Everyone start- ing study of calculu ...

390 Cones and conics 32 This long problem should be very easy. The problem is to read about graphs of equations of the form (1) ...

6.4 Hyperbolas 391 can help usremember, the polar line of P, is the line L, containing thetwo points Q, and R,. A standard simpl ...

392 Cones and conics Multiplying (5) and (6) by (1 - X) and X, respectively, and adding give the relation (9) .4x1x + Byiy = 1, ...

6.5 Translation and rotation of axes 393 or, equivalently, (6.512) x=x'+5, the equation takes the simpler form y = y' + 4, (6.51 ...

394 Figure 6.551 Cones and conics (6.54), we would use the formulas (2.67) (with xo = yo = zo = 0) from Section 2.6 instead of t ...

6.5 Translation and rotation of axes 395 and suppose that .4, B, C are not all zero. Rotating the x, y axes through the angle 0 ...

396 Cones and conies (6.572) to get the information. It is of interest to observe that wecan get information about Q without mak ...

6.5 Translation and rotation of axes 397 of the focus F and the primed equation of the directrix D of the parabola. Finally, fin ...

398 Cones and conics the graph must be a conic. While it is possible to foresee some of the resultsof putting (2) into standard ...

6.5 Translation and rotation of axes 399 have hyperbolic type? Parabolic type? Elliptic type? Hint: Get on the right track. 13 W ...

¢00 Cones and conics Remark: Moving coordinate systems and vectors provide the simplest way of obtaining neat and correct answer ...

6.5 Translation and rotation of axes 401 and (3) O'P = c Icosa b b wti + sin a b b wtj 1 at time t. From (1) and (3) we obtain ( ...

402 Cones and conics the case in which c = b and b = a/4, we obtain the hypocycloid of four having the equation (2) r = 4 [3 cos ...

6.6 Quadric surfaces 403 are notrequired to read it. As is easy to see by considering such special examples as (6.61) x2+y2+z2+1 ...

404 Cones and conics z = k in a set which is the empty set ifjkI > c, a point if Ikl = c, andan ellipse (or circle if a = b) ...

6.6 Quadric surfaces 405 Figure 6.641 The graph of the equation (6.65) a2 --b2-z2=-1 or Figure 6.642 x2 y2 z2 -a2-b2+c2=1 is a h ...