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

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ANALYTIC GEOMETRY

origin in each case being at an arbitrary point (xo, y,). Check
the results by the use of formulas (3) and (4) of the text.
5. y=90~. 6. y = 180. 7. y- -90o.
8. Find the coordinates of the points (0, 0), (1, 2), (- 3, 4),
(- 2, - 5), referred to new axes passing through the point (2, 1),
the angle from the old axis of x to the new being 45~.
9. The same, if the new origin is at the point (-3, 4) and
the angle from the old axis of x to the new is - 30~.
Identify the curve represented by each of the following
equations by transforming to parallel axes at the point (x0, yo)
specified, and then rotating the new axes through the given
angle y. Draw a figure in each case.
Equation (Xo, Yo)
10. 5x2_-6xy+5y2-4x-4y-4=0, (1,1), 45~.
11. x - 4xy y2 + 10x- 2y + 7= 0, (1, 3), -45~.
12. x2 - xy + y' + 46 x + 10y - 47 = 0, (2, 5), 135~.
13. Find the curve represented by the equation
66 2 - 24 xy + 59y2 + 108x + 94y + 76 = 0
by introducing parallel axes at the point (-1, - 1) and then
rotating these axes through the acute angle whose tangent is 4.
Draw a graph.
14. The equation
7x2 - 18xy - 17y2 - 28x+ 36y + 8 =0
represents a conic whose center is at the point (2, 0), and one
of whose axes has the slope —. Identify the conic, and
draw a rough graph of it.
4. Determination of the Transformation from the Equations
of the New Axes. Consider the general transformation given
by formulas (4) of the preceding paragraph. If, in these formulas, we set x'= 0 and y' = 0, we obtain
x cos y + ysin y -- =0,
- x siny - y cosy - y - 0.