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

THE PARABOLA

95

EXERCISES
1. Find the equation of the tangent to the parabola y2 = 3x
at the point (12, 6). Ans. x - 4y + 12 = 0.
2. Find the equation of the normal to the same parabola
at the given point. Ans. 4x + y = 54.
3. Find the length of the latus rectum of the parabola of
Ex. 1.
4. Show that the tangents to any parabola at the extremities of the latus rectum are perpendicular to each other.
5. Show that the tangent to the parabola y2 = 4x at the
point (36, 12) cuts the negative axis of x at a point whose distance from the origin is 36.
6. At what point of the parabola of Ex. 5 is the tangent
perpendicular to the tangent mentioned in that exercise?
Ans. (31, -)7. Show that the two tangents mentioned in Exs. 5 and
6 intersect on the directrix, and that the chord of contact of
these tangents, i.e. the right line drawn through the two points
of tangency, passes through the focus.
8. Show that the tangent to the parabola (1) at any point
P cuts the negative axis of x at a point M whose distance
from the origin is the same as the distance of P from the axis
of y.
9. Prove that the two parabolas,
y2=4x+4 and y2 =- 6x 9,
intersect at right angles. Assume that the slope of the parabola of Ex. 2, ~ 2, at the point (x1, Y) is m/yi.
10. If two parabolas have a common focus and their axes
lie along the same straight line, their vertices, however, being
on opposite sides of the focus, show that the curves cut each
other at right angles.
4. Optical Property of the Parabola. If a polished reflector,
like the reflector of the headlight of a locomotive or a search