By Elliott Mendelson
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Additional resources for Schaum's Outline of Beginning Calculus
21 Find the standard equation of the circle through (−2, 1) and tangent to the line 3x − 2y = 6 at the point (4, 3). 22 Find the standard equation of the circle through (1, 4) and (3, 2) and having its center on the line y = 2x − 1. 1 SLOPE If P1 (x1 , y1 ) and P2 (x2 , y2 ) are two points on a line L , the number m deﬁned by the equation m= y2 − y1 x2 − x 1 is called the slope of L . The slope measures the “steepness” of L . It is the ratio of the change y2 −y1 in the y-coordinate to the change x2 − x1 in the x-coordinate.
By the distance formula, Fig. 4-12 OQ = (1 − 0)2 + (m2 − 0)2 = 1 + m22 OR = (1 − 0)2 + (m1 − 0)2 = 1 + m12 QR = (1 − 1)2 + (m2 − m1 )2 = (m2 − m1 )2 By the Pythagorean theorem for the right triangle QOR, 2 2 QR = OQ + OR 2 (m2 − m1 )2 = (1 + m22 ) + (1 + m12 ) m22 − 2m2 m1 + m12 = 2 + m12 + m22 −2m1 m2 = 2 m1 m2 = −1 Conversely, assume that m1 m2 = −1, where m1 and m2 are the slopes of lines L1 and L2 . Then L1 is not parallel to L2 . ) Let L1 intersect L2 at 34 STRAIGHT LINES [CHAP. 4 point P (see Fig.
22 Find the standard equation of the circle through (1, 4) and (3, 2) and having its center on the line y = 2x − 1. 1 SLOPE If P1 (x1 , y1 ) and P2 (x2 , y2 ) are two points on a line L , the number m deﬁned by the equation m= y2 − y1 x2 − x 1 is called the slope of L . The slope measures the “steepness” of L . It is the ratio of the change y2 −y1 in the y-coordinate to the change x2 − x1 in the x-coordinate. This is equal to the ratio RP2 /P1 R in Fig. 4-1(a). Fig. 4-1 Notice that the value m of the slope does not depend on the pair of points P1 , P2 selected.
Schaum's Outline of Beginning Calculus by Elliott Mendelson