A tangent to the Ellipse makes equal angles with the (219) lines joining the foci and the point of tangency.

Let P be any point of the ellipse, through which the line P'PR is drawn so as to make the angles FPR, F'PP', equal; then will P'PR be tangent at the point P. For, producing F'P to Q, and making PQPF, we have

FP+ P'Q> FP+ PQ = F'P+PF.

Fig. 44.

Now, if P'PR be not a tangent, let the second point, in which it cuts the curve, be P', which we are at liberty to suppose, since P' may be any point of P'PR; then the definition of the ellipse gives F'P' + P'F = F'P+ PF, which is less than F'P' + P'Q;

.. P'F<P'Q, and ... / P'PF < P'PQ, or FPR > QPR = F'PP', which is contrary to the hypothesis; hence, so long as the FPR = F'PP', the line PPR cannot cut the ellipse, and is tangent to it. Q. E. D.

=

Scholium. It is on this account that F, F', are called the Foci of the ellipse; since, from the principle of light and heat, that the angle of reflection is equal to the angle of incidence, if the curve were a polished metallic hoop, and a flame placed at F, the rays, reflected from all points of the ellipse, would pass through F'.

Cor. 1. The tangents drawn through opposite extremities (220) of any diameter, are parallel.

[See (206), (99).]

Fig. 442.

NORMAL TO ELLIPSE.

119

Cor. 2. A parallelogram circumscribing an ellipse will be (221) formed by drawing tangents through the opposite extremities of any two diameters.

Cor. 3. The tangents drawn through the extremities of (222) the axes are at right angles to them, and the circumscribing figure becomes a rectangle.

Fig. 44 3.

Cor. 4. The Normal, or line drawn through the point of tangency perpendicular to the tangent, bisects the angle embraced by the lines drawn from the point of tangency to the foci.

(223)

Fig. 444.

Cor. 5. The axes are normal to the tangents drawn through (224) their extremities.

PROPOSITION III.

To find where the normal intersects the axis of abscissas.

Let TX, n be the normal, intersecting the axis

of X in X,; from (142) we have the proportion

c2

We observe that when c becomes = 0, x, [=2•]

becomes =

a2

0, and .. the normal passes through the centre, O, as it ought to do, since the ellipse becomes then a circle described with

We observe here that the subtangent is independent of the value of b; therefore,

Cor. If upon the same major axis, any number of ellipses (228) of different breadths, and also a circle be described,

then their tangents, drawn to the same abscissa will intersect the axis of x in the same point.

Fig. 453.

EXERCISES.

10. Prove that if two points of a straight line glide along two other lines intersecting at right angles, any third point of the first line will describe an ellipse.

2o. The equation of an ellipse referred to rectangular coordinates is

9y2+4x2 = 36.

Required the distance from the origin to the point in which the normal cuts the axis of x, the abscissa of the point of tangency being 1 = 1.

3°. Where does the normal in 2° cut the axis of y?

4o. Required the subtangent in 2°.

5°. Required the length of tangent in 20 intercepted by the axes

of coördinates.

6°. How far distant from the centre are the foci in 2° ?

7°. What is the eccentricity in 2° ?

8°. Required the parameter in 2°.

9. It is required in 2° to transfer the origin to a point in the ellipse, the abscissa of which shall be x = 162, the new axes being parallel to the old.

This will be done by substituting for y and x, y + y1 and x + x1 =x+12, and observing that

2

9y12 + 4x12 = 36, or 9y,2+4. 1622 = 36.

10°. Given 9y2-90y+4x2+56x +3850, the equation of a

curve referred to rectangular coordinates; it is required to ascertain whether the curve be an ellipse.

Substitute for y, y+y, and for x, x + x1; find the coëfficients of the first power of y and the first power of x in the new equation, and put these coëfficients separately = 0, from which deduce the values of y1, 1. The resulting equation will be found to be 9y2+4x2 = 36.

11°. According to Sir John F. W. Herschel, the equatorial diameter of the earth is 7925 648 miles, and its polar diameter 7899 170 miles. The situation of a place in north latitude is such that a perpendicular dropped upon the earth's axis will intersect it at the distance of 2456 miles from the centre. Required the point in which the direction of a plumb line suspended at the place, will cut the axis of the earth; the meridian being regarded as an ellipse and the plumb as perpendicular to the surface of still water.

SECTION THIRD.

The Hyperbola.

Definition. A Hyperbola is a plane curve described by the intersection of two radii varying in such manner as to preserve in difference the same constant quantity, while they revolve about two fixed points as centres.

Ordering all things as for the equation of the el

lipse, except that the radii are to be denoted by u+a,

X

Fig. 46.

(229)

for the equation of the hyperbola.

The properties of the hyperbola are obviously analogous to those of the ellipse. The student will exercise himself in ascertaining them.

= a;

It has been remarked that the equation of the ellipse, a2y2 + b2x2 =a2b2, becomes that of the circle, y2 + x2 = a2, by making b so the equation of the hyperbola becomes y2 - r2 = — a2, by putting b = a, an equation much resembling that of the circle; hence, the curve which it represents, the Equilateral Hyperbola, possesses properties analogous to those of the circle.

SECTION FOURTH.

The Parabola.

Definition. The Parabola is a plane curve, such that any one of its points is equally distant from a fixed point and a line given in position, which line is denominated the Directrix.

PROPOSITION I.

To find the Equation of the Parabola. Take the directrix, D, for the axis of y, and for the axis of the perpendicular to D, drawn through the fixed point, F, whose distance from D we will indicate by p. Then, by the definition, P representing any point of the curve, we have y2+(px) = x2, y2 = 2px - p2,

is an equation of the parabola.

=

O for the purpose of finding

If in this equation we make y where the curve cuts the axis of x, there results

Cor. 1. The Parabola cuts the axis of x midway be- (231)

tween the fixed point F and the directrix.

In order to transport the origin to this point, we have only to substitute in (230) for x, x+p; doing which, there results

y2 = 2px,

Fig. 47 3.1

(232)

the equation of the parabola.

Here we observe that as 'increases y increases, and that without limit, and for every value of x there results two equal values of y; also, x does not admit of any minus value, since in that case y, [ = ( − x)], would be imaginary; hence,