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

72

ANALYTIC GEOMETRY

7. 2x2 + 2y2-3x -y= 1 at (-1,2).
8. Find the intercepts on the axis of x made by the tangent
at (- 5, 12) to
x2 + y2 = 169. Ans. -33.
9. Find the area of the triangle cut from the first quadrant
by the tangent at (1, 1) to
32 + 3y2 + 8x + 16y = 30.
10. If the equation
x2 + y2 + Ax - By + C= O
represents a circle, and if the point (x1, yj) lies on the circle,
show that the equation of the tangent at this point can be
written in the form:
(9) xix + lyy (x + i)(+ (y I Yi) + C= 0.
Suggestion. Find the values of a, /3, and p for the circle,
substitute them in (6), and simplify the result.
11. Do Exs. 6 and 7, using formula (9), Ex. 10.
12. The same for the tangent to the circle in Ex. 9.
13. Show that, if PI: (x1, y') is any point of the circle
x2 + y2 + Ax + By+ C= 0,
at which the tangent is not parallel to the axis of y, then the
slope of the tangent at Pi is
2x1 + A
2y +- B
4. Circle through Three Points. It is shown in Elementary
Geometry that a circle can be passed through any three points
not lying in a straight line.
If the points are (x1, Yi), (x2, y2), and (X3, y3), and if the
equation of the circle through them is written in the form
2 + y2 + Ax + By+ C= 0,
then clearly the following three equations must hold: