177. The sum of the squares of any two conjugate diameters of an ellipse being constant, their rectangle is a maximum when they are equal; and, therefore, in this case, sino is a minimum; hence the acute angle between the two equal conjugate diameters is less (and, consequently, the obtuse angle greater) than the angle between any other pair of conjugate diameters.

: The length of the equal conjugate diameters is found by making a'b' in the equation a+b2 = a + b2, whence a" is half the sum of a2 and b2, and in this case

The angle which either of the equiconjugate diameters makes with the axis of x is found from the equation

by making tan-tane'; for any two equal diameters make equal angles with the axis of x on opposite sides of it (Art. 162). Hence

b

a

It follows, therefore, from Art. 167, that if an ellipse and hyperbola have the same axes in magnitude and position, then the asymptotes of the hyperbola will coincide with the equiconjugate diameters of the ellipse.

The general equation of an ellipse, referred to two conjugate diameters (Art. 168), becomes x2+ y2=a"2, when a'=b'. We see, therefore, that, by taking the equiconjugate diameters for axes, the equation of any ellipse may be put into the same form as the equation of the circle, x2 + y2 = r2, but that in the case of the ellipse the angle between these axes will be oblique.

178. To express the perpendicular from the centre on the tangent in terms of the angles which it makes with the axes.

If we proceed to throw the equation of the tangent

yy' + =1) into the form x cosa+y sina=p (Art. 23),

b2

we find immediately, by comparing these equations,

Substituting in the equation of the curve the values of x', y', hence obtained, we find

p*=a* cos2a+b2 sin3a.*

The equation of the tangent may, therefore, be written

x cosa+y sina-√(a cosa + b2 sin2 a) = 0.

Hence, by Art. 34, the perpendicular from any point (x'y') on the tangent is

√(a2 cos2a+b2 sin' a) — x cosa -y' sina,

where we have written the formula so that the perpendiculars shall be positive when x'y' is on the same side of the tangent as the centre.

Ex. To find the locus of the intersection of tangents which cut at right angles. Let p, p' be the perpendiculars on those tangents, then

p2 = a2 cos2 a + b2 sin2a, p'2 = a2 sin2 a + b2 cos2a, p2+p22 = a2 + b2.

But the square of the distance from the centre, of the intersection of two lines which cut at right angles, is equal to the sum of the squares of its distances from the lines themselves. The distance, therefore, is constant, and the required locus is a circle (see p. 161, Ex. 4).

179. The chords which join the extremities of any diameter to any point on the curve are called supplemental chords.

Diameters parallel to any pair of supplemental chords are conjugate.

For if we consider the triangle formed by joining the extremities of any diameter AB to any point on the curve D; since, by elementary geometry, the line joining the middle points of two sides must be parallel to the third, the diameter bisecting AD will be parallel to BD, and the diameter bisecting BD will be parallel to AD. The same thing may be proved analytically, by forming the equations of AD and BD, and showing that the product of the tangents of the angles made by these lines with b2

the axis is =

a2.

This property enables us to draw geometrically a pair of conjugate diameters making any angle with each other. For if we describe on any diameter a segment of a circle, containing the

* In like manner, p2 = a'2 cos2 a + b'2 cos2 ß, a and ß being the angles the perpendicular makes with any pair of conjugate diameters.

given angle, and join the points where it meets the curve to the extremities of the assumed diameter, we obtain a pair of supplemental chords inclined at the given angle, the diameters parallel to which will be conjugate to each other.

Ex. 1. Tangents at the extremities of any diameter are parallel.

This also follows from the first theorem of Art. 146, and from considering that the centre is the pole of the line at infinity (Art. 154).

Ex. 2. If any variable tangent to a central conic section meet two fixed parallel tangents, it will intercept portions on them, whose rectangle is constant, and equal to the square of the semi-diameter parallel to them.

Let us take for axes the diameter parallel to the tangents and its conjugate, then the equations of the curve and of the variable tangent will be

x2 y2
+
a'2 6'2=1,

The intercepts on the fixed tangents are found by making x alternately = ± a' in the latter equation, and we get

which, substituting for y'2 from the equation of the curve, reduces to b'2.

Ex. 3. The same construction remaining, the rectangle under the segments of the variable tangent is equal to the square of the semi-diameter parallel to it.

For, the intercept on either of the parallel tangents is to the adjacent segment of the variable tangent as the parallel semi-diameters (Art. 149); therefore, the rectangle under the intercepts of the fixed tangents is to the rectangle under the segments of the variable tangent as the squares of these semi-diameters; and, since the first rectangle is equal to the square of the semi-diameter parallel to it, the second rectangle must be equal to the square of the semi-diameter parallel to it.

Ex. 4. If any tangent meet any two conjugate diameters, the rectangle under its segments is equal to the square of the parallel semi-diameter.

Take for axes the semi-diameter parallel to the tangent and its conjugate; then the equations of any two conjugate diameters being (Art. 170)

the intercepts made by them on the tangent are found, by making x = a', to be

We might, in like manner, have given a purely algebraical proof of Ex. 3. Hence, also, if the centre be joined to the points where two parallel tangents meet any tangent, the joining lines will be conjugate diameters.

Ex. 5. Given, in magnitude and position, two conjugate semi-diameters, Oa, Ob, of a central conic, to determine the axes.

P

M

The following construction is founded on the theorem proved in the last Example:-Through a, the extremity of either diameter, draw a parallel to the other; it must of course be a tangent to the curve. Now, on Oa take a point P, such that the rectangle Oa.aP Ob2 (on the side remote from O for the ellipse, on the same side for the hyperbola), and describe a circle through O, P, having its centre on aC, then the lines OA, OB are the axes of the curve; for, since the rectangle Aa.aB = Oa.aP = Ob2, the lines OA, OB are conjugate diameters, and since AB is a diameter of the circle, the angle AOB is right.

Ex. 6. Given any two semi-diameters, if from the extremity of each an ordinate be drawn to the other, the triangles so formed will be equal in area.

Ex. 7. Or if tangents be drawn at the extremity of each, the triangles so formed will be equal in area.

THE NORMAL.

180. A line drawn through any point of a curve perpendicular to the tangent at that point is called the Normal.

Forming, by Art. 32, the equation of a line drawn through 'xx, yy' (x'y') perpendicular to + a" b2

of the normal to a conic

or

=1), we find for the equation

c2 being used, as in Art. 161, to denote a2 — b2.

Hence we can find the portion CN intercepted by the normal

on either axis; for, making y=0 in

the equation just given, we find

B

for given CN we can find x', the abscissa of the point through which the normal is drawn.

=

=

The circle may be considered as an ellipse whose eccentricity =0, since ca2-b0. The intercept CN, therefore, is constantly = 0 in the case of the circle, or every normal to a circle passes through its centre.

181. The portion MN intercepted on the axis between the normal and ordinate is called the Subnormal. Its length is, by

The normal, therefore, cuts the abscissa into parts which are in a constant ratio.

If a tangent drawn at the point P cut the axis in T, the intercept MT is, in like manner, called the Subtangent.

The length of the normal can also be easily found. For

But if b' be the semi-diameter conjugate to CP, the quantity within the parentheses = b2 (Art. 173). Hence the length of the

bb'

a

If the normal be produced to meet the axis minor, it can be

proved, in like manner, that its length

==

ab'

b

Hence, the rect

angle under the segments of the normal is equal to the square of the conjugate semi-diameter.

Again, we found (Art. 175) that the perpendicular from the

centre on the tangent

=

ab
b'

Hence, the rectangle under the

normal and the perpendicular from the centre on the tangent, is constant and equal to the square of the semi-axis minor.

Thus, too, we can express the normal in terms of the angle it makes with the axis, for

Ex. 1. To draw a normal to an ellipse or hyperbola passing through a given point. The equation of the normal, a2xy' - b2x'y = c2x'y', expresses a relation between the co-ordinates x'y' of any point on the curve, and xy the co-ordinates of any point on the normal at x'y'. We express that the point on the normal is known, and the point on the curve sought, by removing the accents from the co-ordinates of the latter

Ꮓ