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

98

ANALYTIC GEOMETRY

7. A circle is tangent to the parabola y2 = x at the point
(4, 2) and goes through the vertex of the parabola. Find its
equation.
8. What is the equation of the circle which is tangent to
the parabola y2 = 2 mx at both extremities of the latus rectum?
Ans. 4X2 + 4y2 - 12mx + m2 = 0.
9. Find the coordinates of the points of tangency of the
tangents to the parabola y2 = 2 mx which make the angles 60~,
45~, and 30~ with the axis of the parabola. Show that the
abscissae of the three points are in geometric progression, and
that this is true also of the ordinates.
10. Show that the common chord of a parabola, and the
circle whose center is in the vertex of the parabola and whose
radius is equal to three halves the distance from the vertex to
the focus, bisects the line-segment joining the vertex with the
focus.
11. Let N be the point in which the normal to a parabola
at a point P, not the vertex, meets the axis. Prove that the
projection on the axis of the line-segment PN is equal to one
half the length of the latus rectum.
12. On a parabola, P is any point other than the vertex,
and N is the point in which the normal at P meets the axis.
Show that P and N are equally distant from the focus.
13. The tangent to a parabola at a point P, not the vertex,
meets the directrix in the point L. Prove that the segment
LP subtends a right angle at the focus.
14. Show that the length of a focal chord of the parabola
y2 = 2 mx is equal to x1 + x2 + m, where x1, x2 are the abscissae
of the end-points of the chord. Hence show that the midpoint of a focal chord is at the same distance from the directrix as it is from the end-points of the chord.
Exercises 15-26. The following exercises express properties
of the parabola which involve an arbitrary point on the parabola. In order to prove these properties, it will, in general, be