14 Analytic geometry in two dimensions
slope m of L, to find the equation of L, and to determine xo algebraically. Do all
this and make the results agree when(a) (x1,Y1) _ (1,1), (x2,Y2) = (3,2)
(b) (xl,Y1) = (L-1), (x2,Y2) = (3,1)
(c) (xl,yi) = (-4,-2), (x2,Y2) = (-1,-1)
(d) (xi,Yi) _ (0,4), (x2,Y2) = (1,2)
(e) (x1,Y1) _ (1,2), (x2,Y2) = (2,1)
(1) (x1,Yi) = (-1,1), (x2,Y2) = (4,-1)11 Plot at least five points P(x,y) whose coordinates satisfy the equation
y = 2x - 4. The coordinates can be found by giving values such as -1, 0, i,
1 to x and calculating y. Observe that these points appear to lie on a line L.
Show that the given equation can be written in the form y - 0 = 2(x - 2) and
hence that the points must lie on the line L through the point (2,0) which has
slope 2. Make everything check.
12 Supposing that a and b are nonzero constants, find the point-slope form
of the equation of the line L through the two points (a,0) and (0,b), and show that
this equation can be put in the formsbx+ay-ab=0, 6+b=1.
The second form is the intercept form of the equation of L. Note that it is very
easy to put y = 0 and see that L intersects (or intercepts) the x axis at the point for
which x = a. It is equally easy to put x = 0 and see that L intersects (or inter-
cepts) the y axis at the point for which y = b.
13 A line intersects the x axis at the point (a,0) and cuts from the first quad-
rant a triangular region having area A. Find the equation of the line. Ins.:
24x + a2y = 2a11.
14 For each of the cases(a) P1 = (1,1), P2 = (7,1), Pa =
(b) P1 = (2,2), P2 = (8,2), Ps =(7,7)
(8,8)
(c) P1 = (-3,-1), P2 = (2,-7), P3 = (4,1)
(d) P1 = (-4,-2), P2 = (1,-8), P3 = (3,0)
(e) P1 = (-2,-4), P2 = (1,-5), P3 = (2,-3)sketch the triangle having vertices P1, P2, P3 and the line L containing P, and the
mid-point of the side opposite P1. Use the figure to obtain an estimate of the
x coordinate of the point where L intersects the x axis. Then find the equation
of L and determine the coordinate algebraically. Produce results that have
reasonable agreement. Remark: There is one respect in which many problems
in pure and applied mathematics are like this one. Graphs or something else
give more or less good approximations to answers, but we need equations to get
correct answers. When equations give answers that seem to be wrong, the whole
situation must be given close scrutiny. Mistakes in sign are particularly damag-
ing, and we all make mistakes when we work too rapidly or too thoughtlessly.
15 A triangle with vertices 11, B, C is placed upon a coordinate system in
such a way that A is at the origin and the mid-point D of the opposite side is