Analytic geometry by Lewis Parker Siceloff, George Wentworth and David Eugene PDF
By Lewis Parker Siceloff, George Wentworth and David Eugene Smith
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2. Find the point which divides the line segment from (- 4, - 2) to (1, 3) externally in the ratio 8 : 3. Since x1 =- 4, x2 = 1, and r =X 0 Similarly, -~, -4+(-ll) = " = 1 + (- ~) we have _2Q _-Jr_ -% = 4. - - 2 + (- ~). 3- -10 Yo- _ § -6. ) Hence the required point is (4, 6). DIVISION OF A LINE SEGMENT 25 Exercise 8. Points of Division Determine, without writing, the mid point of each qf these line segments, the end points being as follows : 4. (4, -1), (- 4, 1). ,_ 5. (a, 1), (1, a). 1. (7, 4), (3, 2).
In a given function such as f ( x) we write f (a) to mean the result of substituting a for x in the function. · For example, if f(x) = x~- ~' then f(1) = 1-1 = 0, j(3) = 8~. Exercise 18. Functions and Graphs 1. f(3), and f(4). 2. ), and f(5). 3. Draw' the graph showing the relation between t and 1n as given in the table in § G1, page 55. 4. 7 5. 49 meaning 4 hr. 49 min. 4i'i Can points on the graph be found approximately for d = 10? for What kind of number must d be? Should the graph be a continuous curve?
It is therefore evident that all points (x, y) whose coordinates satisfy such an equation are precisely all the points whose coordinates make one or more factors 0. In other words, The graph of a degenerate equaf'ion consists of the graphs qf the several equations obtained by placing equal to 0 the several factors that contain either x or y, or both x and y. For example, the graph of the equation x 2 - xy- 3 y + 3 x = 0, which may be written (x- y) (x + 3) = 0, consists of the two lines x - y = 0 and x + 3 = 0.