A Course in Mathematics, for Students of Engineering and Applied Science Volume N . 1 Book PDF, ePub eBook

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Book Descriptions:
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. 1907 Excerpt: ... suggests a method of finding the tangent to any curve represented by an equation of the second degree, the slope of the tangent being given. For if m of the required tangent is known, its equation may be written y = mx + b, where b is not known. According to the definition of a tangent, however, b must have such value that the points of intersection of straight line and curve shall be coincident. This condition enables us to determine b, as shown in the following examples. Ex. 1. Find the equation of the tangent to the parabola 3 x2 + 2 y=0, the slope of the tangent being 2. Since the slope of the tangent is 2, its equation may be written y = 2 x + b. Substituting this value of y in the equation of the parabola, we have the equation 3x2 + 4x + 26 = 0. Since the line is to intersect the curve in two coincident points, this equation must have equal roots. The condition for equal roots, by § 37, is (4)2 4 (3) (2 6) = 0, whence we find 6 = 3-. Therefore the required tangent isy = 2x + 5-, or6x--3y + 2 = 0 (fig. 98). Ex. 2. Find the equation of the tangent to the ellipse x2 + 4 y2 = 4, the slope of the tangent being The equation of the tangent is y = x + b. Substituting this value of y in the equation of the ellipse, we have By this same method the following formulas for a tangent with known slope m may be derived: "1. The tangent to the parabola y" = 4px is if 2. The tangents to the ellipse--+ = 1 are 89. It was stated in § 87 that the result of the substitution is in general a quadratic equation. In exceptional cases, however, the resultant equation may be linear, as in the first of the following examples, or even impossible, as in the second example. Ex. 1. Consider 2x-by-10 = 0 (1) and 4x2-25 u2 = 100. (2) Fi;. 100 Substituting in (2) th...

A Course in Mathematics, for Students of Engineering and Applied Science Volume N . 1 by Frederick Shenstone Woods

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