This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ...find its equation. 9. Prove that it follows from Ex. 8 that the centers of all conies through four given points (no three of which are on the same straight line) lie on a conic. 10. Find the two conics of the confocal system x2/(3 + X)-t-j/2/(2--X) = 1 which pass through the point (2, 1). 11. Prove that the equation of ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ...find its equation. 9. Prove that it follows from Ex. 8 that the centers of all conies through four given points (no three of which are on the same straight line) lie on a conic. 10. Find the two conics of the confocal system x2/(3 + X)-t-j/2/(2--X) = 1 which pass through the point (2, 1). 11. Prove that the equation of the hyperbola confocal to the ellipse x2/a2 + y2/b2 = 1 and meeting the ellipse at the point whose eccentric angle is 0 is x2/cos2 j, --y2/sXn2 $ = a2--62. If 2 gx' + 2fy' + c be added to both members of this equation, the right member will vanish, since /(a;', y') = 0, and the equation, after dividing by 2, will become axx' + h(xy' + yx') + byy' +g(x + x') +f(y + y')+c = 0. (3) Hence, to obtain the equation of the tangent at the point (x', y') from the equation of the curve, it is only necessary to replace x2 and y2 by xx' and yy', 2 xy by xy' + x'y, and 2 x and 2 y by a; + a;' and?/ + y'. (This is true for oblique axes also.) Thus, the equation of the tangent at (x', y') to the curve 2x2-5zy + t/2 + 4a;-3!/ + 7= 0 is 2 xx'-f (ay' + x'y)+yy' + 2(x + z')-l(y + y')+ 7 = 0.( 172. Poles and polars. The equation (3) of 171 represents a tangent to the conic (1) only when the poin.t (a;', y') is on the conic. But, whether the point lies on We conic or not, the equation represents a definite straight line. This line is called the polar of the point (x', y') with respect to the conic (1), and (x', y') is called the pole of the line. From the symmetry of the equation (3) with respect to x, y on the one hand, and x', y' on the other hand, and the fact that (3) represents the tangent at (-', y') when (x', y') is on the curve, it is not difficult to infer the geometric relation between any point (x', y') not on the curve and its polar (3). Onl...
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