Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

234

ANALYTIC GEOMETRY

GENERAL TRANSFORMATION OF AXES
28. Prove that the straight lines
3x — 4y —2=0, x+ 2y =4,
when referred to suitable axes, will have equations the first
of which is x' = 0, while the second contains no constant term.
29. Find the equations of the circle
2 + y2 + 6 - 8y + 6=0
and the line 5x + 12y - 13 = 0,
when they are referred to axes through the center of the circle
parallel and perpendicular, respectively, to the line.
30. Find the equations of the two circles
x2 + y2+ 4x - 6y - 4 = 0,
x2 + y2 6x + 4y - 4 = 0,
when they are referred to the point midway between their
points of intersection as origin and the line joining the points
of intersection as axis of x'.
31. What will the equations of the circles of Ex. 30 become,
if they are referred to the mid-point between their centers as
origin and the line of the centers as axis of y'?
32. A transformation consists of a change of origin to the
point (x0, ye), of a rotation of the new axes through the angle
y, and of a reversal of the sense of the axis of x thus obtained.
Show that the equations of the transformations are
x =- -x' cos y - y' sin y + x0,
y = - x' siny + y' cos 7 + y0.