18 Analytic geometry in two dimensions
two points. Then find in by putting the equation in the form y = mx + b
Make the results agree.
(a) x + y = 2 (b) x + y = 3
(c) 2x+y=2 (d) x+2y=2
(e) 2x+3y-40 (f) 2x+3y+40
(g) 2x-3y-4=0 (h) 2x-3y+40
(i) 1+2=1 (1) 1-2=12 Draw the triangle having vertices at the points P1(-3,1),P2(7,-1),
P3(1,5), and observe that PsP2 seems to be nearly perpendicular to P1P3. Find
the equation of the line through P3 perpendicular to PiP3 and show that this line
does contain the point F. (The figure appears among the problems at the end
of Section 1.2.)
3 For the points P1, Ps, P3 of the preceding problem, find the equation of
the line through P2 parallel to the line P1P3. Find the coordinates of the point
in which this new line intersects the y axis. Put this new line into the figure,
and make everything check.
4 For each of the following equations find numerical coordinates of three
points P1, P2, P3 whose coordinates satisfy the equation. If you cannot think
of a better procedure, let x1 = 0, x2 = 1, x3 = 2 and calculate y1i y2, ys Plot
the three points P1, P2, P3 and notice that they seem to lie on a line. Calculate
the Slopes of P2P2 and P1P3 and observe that they are equal. Observe that there
is ample opportunity to check all answers.(a) y=x+1 (b) y=2x+3
(c) x+y=S (d) x + y + 2 = 0
(e) 2x-3y+4=0 (f) 2x+3y+4=0
5 Plot the lines having the equations y = 2x and y = 3x and observe that
the acute angle 0 between them seems to be rather small. Find 0 by finding
tan ,, and then construct and examine an appropriate figure to see that your
answer seems to be correct.
6 Supposing that k is a nonnegative number, find the acute angle between
the lines having the equations y = kx and y = (k + 1)x. Check the answer in
at least one special case. Tell why the angle should be small when k is large.
7 Sketch the line L1 which intersects the coordinate axes at the points (0, -4)
and (5,0), and the line L2, which intersects the coordinate axes at the points
(0, - 5) and (6,0). Find the acute angle between the lines. ins.: tan 0 = ,'T.
8 While assembly lines and mass production reduce costs of manufactured
items, there is an element of sanity in the idea that the total cost y of publishing
x copies of a book is ax + b. Sketch a graph of the equation y = ax + b and
discover the significance of the numbers a and b.(^9) Find the equation of the perpendicular bisector of the line segment joining
the points P1(x1,y1) and Ps(x2,y2), putting the answer in the form Ax + By = C.
ins.:
z z z 2
(x2 - x1)x + (ys - yi)Y=xz
2
xl + Yz
2
Y1,