Calculus: Analytic Geometry and Calculus, with Vectors

Cones and conics

8 Let P, be a point on a parabola which is not the vertex P. Let W be the

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intersection of the tangents to the parabola at Pl and V. Show that the line

Figure 6.192

from the focus F to W is perpendicular

to the line WP1. Hint: Let the parabola

have the equation y = kx2 and use the fact

that F has coordinates (0, 1/4k). See

Figure 6.192.

9 Let Pl be a point on a parabola which

is not the vertex V. Prove that the tangent

to the parabola at Pl meets the directrix and

the line through the focus F parallel to the

directrix at two points Q and R that are

equidistant from F. See Figure 6.192.

10 A particle P moves on the parabola

having the equation y = kx2 in such a way

that, at each time t, its x and y coor-

dinates are t and kt2 and the vector r running from the origin to P is

r = ti + kt2j.

Show that the velocity vector v is

v=i+2ktj,

and note that this vector is also a "fore and tangent" to the parabola at P.

Letting F be the focus of the parabola, show that

FP=ti+(kt'--k}j and

Letting cal be the angle between the vector FP and the tangent vector v, show

that

2ki

cos ¢i =

1,1 + 4k2t2

Letting 452 be the angle between the tangent vector v and the vertical vector j,

show that

cos 02 =

IVI IjI -\/1+t4k2t2

and hence that 02 = 01. Remark: These formulas yield the famous reflection

property of parabolas. They imply that the line FP and the line extending

upward from P make equal angles with the normal to the parabola at P. This

implies that if light or something else goes in a line from F and is reflected from

the parabola in such a way that the angle 0,. of reflection is equal to the angle

0, of incidence, then its path after reflection is parallel to the axis of the parabola.

(^11) Modify the work of the preceding problem to make a direct attack upon
the angles which FP and j make with the normal to the parabola at P. Remark:
This should be done when our primary interest lies in angles of incidence and
reflection.