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

THE ELLIPSE 113
4. At what angle does the straight line through the origin,
which bisects the angle between the positive axes of coordinates,
cut the ellipse 3x2 + 4y2 = 7? Ans. 81~ 53'.
5. Find the area of the triangle cut off from the first quadrant by the tangent to the ellipse of Ex. 3 at the point (1, 2).
Ans. 8-.
6. Find the length of the latus rectum of the ellipse of
Ex. 1. Ans. 3.
7. The same for the ellipse of Ex. 3.
8. Show that the length of the latus rectum of the ellipse
+ y 1, b < a,
at b2
is given by any one of the expressions
2 b2
2 —; 2b1 - e2; 2a( - e2).
Find its value in terms of c and e.
9. Find the length of the latus rectum of the ellipse
25 x2 + 16 y2= 400. Ans. 62.
10. Prove that the minor axis of an ellipse is a mean proportional between the major axis and the latus rectum.
7. A New Locus Problem. Given a line D and a point F
distant m from D. To find the locus of a point P such that
the ratio of its distance FP from F to y
its distance MP from D is always /
equal to a given number, E:
FP
(1) P = or FP= eEMP.
0MP i F(mz,0)
It is understood that P shall be restricted to the plane determined by F \
and D.
If, in particular, =1, the locus FIG. 11