Calculus: Analytic Geometry and Calculus, with Vectors

6.1 Parabolas 359

12 This problem and Figure 6.193

delve a bit deeper into the geometry of

parabolas. The figure shows the pa-

rabola having the equation y = kx2,

where k > 0. The focus F and the

directrix have the coordinates (0, 1/4k)

and the equation y = -1/4k. Let

x, > 0. Show that the line through

P,(x,, kxi) parallel to the axis of the pa-

rabola intersects the directrix at the

point D,(x,, -1/4k). Show that the

tangent to the parabola atP, intersects

the axis of the parabola at the point

Q,(0,-kx2). Show that the quadrilat-

eral Q,D,P1F is a rhombus, that is, an

equilateral parallelogram. This rhom-

bus could be called a focal rhombus; in

any case itis a focal square when

x, = 1/2k and the rhombus is a square.

Figure 6.193

Show that the diagonals of this rhombus

are perpendicular to each other and that they intersect at the point (x,/2, 0).

Finally, show how these results and elementary geometry can be used to prove

the reflection property of the parabola, namely, that the line from the focus to

P1 and the line through P, parallel to the axis of the parabola make equal angles

with the tangent to the parabola at P,.

13 Write the equation of the tangent to the graph of the equation y = x2

at the point (x,,x1) and then try to determine x1 so the tangent will contain (or

pass through) the point

(a) (1,1) (b) (1,0) (c) (0,1) (d) (-1,-1).

(^14) Find the equation of the normal to the graph of y = x2 at the point (x,,xi)
on the graph. Show that if yo 5 -, then there is only one value of x, for which
the normal passes through the point (O,yo), but that if yo > , then there are
three values of x1 for which the normal passes through the point (O,yo). Sketch
a figure or figures which show that the results seem to be reasonable.
15 As is the case for circles, a line segment joining two points on a parabola
is a chord of the parabola, and the set of mid-points of the chords parallel to a
given chord is called a diameter of the pa-
rabola. Thus a diameter is a point set, not
a number. Letting the parabola have the
equation y = kx2, where k > 0, prove that
for each in the diameter determined by
chords having slope in is the line segment
containing points (x,y) for which x = m/2k
and y > m2/4k.
16 A focal chord of a parabola is a line
segment which contains the focus and has its
ends at points on the parabola. Supposing
that x2 < 0 < x, as in Figure 6.194, show
Figure 6.194