By Lewis Parker Siceloff, George Wentworth and David Eugene Smith
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32 GEOMETRIC MAGNITUDES 61. Given that AP = PB, AP is perpendicular to PB, A is (2, 3), and B is (- 3, - 2), find the coordinates of P. 62. The mid point of the hypotenuse of a right triangle is equidistant from the three vertices. This theorem, familiar from elementary geometry, makes no mention of axes. ·we, therefore, choose the most convenient axes, which, in this case, are those which lie along OA and OB. From the coordinates of A (a, 0) and of B(O, b), we can find those of M and the three distances referred to in the exercise.
2 2 14. f(x-4)-x2 (x-8)=0. 15. x 2(y+8)+y3 =0. 16. f. 17. 2 :J =9(x2-2x-8). ~-6x+5 18. y = ') x-4 2 """'x- 7 x- 4 19. A and Bare two centers of magnetic attraction 10 units apart, and P is any point of the line AB. P is attracted by the center A with a force P1 equal to 12/A P 2, 10 and by ~e center B with a force F 2 equal ~to 18jBP2• Letting x=AP, express in terms of x the sum s of the two forces, and draw a graph showing the variation of s for all v::Llues of ;::;, 52 LOCI AND THEil~ EQUATIONS 54.
P 0 is on P 1P 2 produced. 46. P 0 is ori P 2P 1 produced. 47. Given the point A (1, 1), find the point B such that the length of AB is 5 and the abscissa of the mid point of AB is 3. 48. In a triangle ABC, if A is the point (4, -1), if the mid point of AB is 1lf (3, 2), and if the medians of the triangle meet at P (4, 2), find C. 49.. Find the point Q which is equidistant from the coordinate axes and is also equidistant from the points A ( 4, 0) and B (- 2, 1). • REVIEW 31 50. Given the points A (2, 3) and B (8, 4), find the points which divide AB in extreme and mean ratio, both internally and externally.
Analytic geometry by Lewis Parker Siceloff, George Wentworth and David Eugene Smith