451. - Ellipse: Definitions. - Let two lines, PF, PF' (Fig. 299), be drawn from any point P to any two fixed points FF', and let the point P move in such a manner that the sum of the two lines, PF, PF', shall remain a constant quantity; then the curve PMKO GA DBP, traced by P, will be an Ellipse; the two fixed points F, F' the Foci; the point C at the middle of FF' , the centre; the line A M drawn through FF' and terminated by the curve, the Major or Transverse Axis; the line B O, drawn through C and at right angles to A M, the Minor or Conjugate Axis; the line G P, drawn through P and C and terminated by the curve, the Diameter to the point P; the line D K drawn through C, parallel with the tangent P T, and terminated by the curve, the diameter Conjugate to PG; the line EHR drawn parallel with DK is a double ordinate to the abscissas GH and HP of the diameter GP (E H = HR); the line JL drawn through F at a right angle to A M and terminated by the curve, the Parameter, or Latus Rectum.
When the point P reaches and coincides with B, the two lines PF and PF' become equal.
The proportion between the major and minor axes depends upon the relative position of F, F', the foci; the nearer these are placed to the extremities of the major axis the smaller will the minor axis be in comparison with the major axis. The nearer F, F' approach C, the centre, the nearer will the minor axis approach the length of the major axis. When F, F' reach and coincide with the centre, the minor axis will equal the major axis, and the ellipse will become a circle. Then we have PF= PF = B C= A C. From this we learn PF+PF'=2AC=AM; also, when PF= PF', then PF=BF= A C.
From this we may, with given major and minor axes, find the position of Fand F'. To do this, on B, as a centre, with A C for radius, mark the major axis at Fand F'.
452. - Ellipse: Equations to the Curve. - An equation to a curve is an expression containing factors two of which, called co-ordinates, measure the distance to any point in the curve. For example: in a circle it has been shown (Art. 443) that PN is a mean proportional to A N and NB. Or, putting x - A N, y = PN, and a - A B, we have -
An: Pn:: Pn: Nb,
or - x: y:: y: a - x,
or - y 2 = x (a - x).
This is the equation to the circle having the origin of x and y, the co-ordinates at A,the vertex of the curve. It will be observed that the factors are of such nature in this equation, that it may be employed to measure the distance, rectangularly, to P wherever in the curve the point P may be located. By this equation the rectangular distance to any and every point in the curve may be measured; or, having the curve and one of the lines x or y, the other may be computed.
From this example, the nature and utility of an equation to any curve may be understood. The equation to the ellipse having the origin of co-ordinates at the vertex, is similar to that for the circle. In the form usually given by writers on Conic Sections, it is -
y2 = b2/a2 (2ax-x2) (173.)
in which a = A C (Fig. 299);b = B C; x equals A N, and y = PN.
If, as before suggested, the foci be drawn towards the centre and finally made to coincide with it, the minor axis would then become equal to the major axis, changing the ellipse into a circle. In this case, the factors a and b in the equation would become equal; and the fraction b2/a2would equala2/a2 = 1,and hence the equation would become -
y2 = 2 ax - x2, or - y2 = x (2 a - x);
precisely the same as in the equation to the circle above shown. The 2 a of this equation is equivalent to a of the circle; for a in the ellipse represents only half the major axis; while in the equation to the circle a represents the diameter. The relation between the ellipse and the circle is thus shown; indeed, the circle has been said to be an ellipse in its extreme conditions.
453. - Ellipse: Relation of Axis to Abscissas of Axes. - Multiplying equation (173.) by a2 we have -
a2 y2 = b2 (2 a x - x2),
or - a2 y2 = b2 x (2 a - x).
These four factors may be put in a proportion, thus -
aa: b2:: x (2 a - x): y2, representing: -
Or: The rectangle of the two parts into which the ordinate divides the axis major is in proportion to the square of the ordinate, as the square of the semi-axis major is to the square of the semi-axis minor.
It is shown by writers on Conic Sections that this relation is found to subsist, not only with the axes and ordinate, but also between an ordinate to any diameter and the abscissas of that diameter; for example, referring to Fig. 299 -
If A B' P'M (Fig. 301) be a semi-circle, then (Art. 443)
Substituting this value of A NxNM in-
we have -
AC: BC:: P'N: PN;
Or: The ordinate in the circle is in proportion to its corresponding ordinate in the ellipse, as the semi-axis major is to the semi-axis minor, or as the axis major is to the axis minor.
454. - Ellipse: Relation of Parameter and Axes. - The equation to the ellipse when the origin of the co-ordinates is at the centre is, as shown by writers on Conic Sections, thus -
a2y2 = a'b'-b'x'2, (174.)
or - a2y2 = b2(a2-x'2).
If x' equal CF(Fig. 299) then the ordinate will be located at FJ, and -
x'2 = a2 - b2 .
Then - a2 - x'2. = a2 - (a2 - b2)
= a2 - a2 + b2,
a2-x'2 = b2
This is shown also by the figure.
Substituting in the above this value of a2 - x'2, we have -
a2y2 = b2 b2 = b4
From which, taking the square root -
ay = b2; or - a: b:: b: y.
Now y, located at FJ, is the semi-parameter; hence we have the semi-minor axis a third proportional to the semi-major axis and the semi-parameter. Or: The parameter is a third proportional to the two axes of an ellipse,
455. - Ellipse: Relation of Tangent to the Axes. - Let
T T1 (Fig. 301) be a tangent to P, a point in the ellipse; then, as has been shown by writers on Conic Sections -
or - CM: CT:: CN: CM.
Or: The semi-major axis is a mean proportional between the abscissa C N and C T, the part of the axis intercepted between the centre and the tangent.
This relation is found also to subsist between the similar parts of the minor axis; for -
This relation affords an easy rule for finding the point T, or T1; for from the above we have -
or, putting t for C T, we have -
t = a2/x1 (175.)
t1 = b2/y. (176.)
Since the value of t is not dependent upon y nor upon b, therefore t is constant for all ellipses which may be described upon the same major axis A M; and since the circle is an ellipse (Art. 452) with equal major and minor axes, therefore rule (175.) is applicable also to a circle, as shown in Fig. 301.
The equation (175.) gives the value of t = C T. From this deducting CN = x', we have N T, the sabtangcnt, or -
t - x' = s;
or, substituting for t its value in (175.), we have -
s = a2/x' - x';
Or: The subtangent to an ellipse equals the difference between the quotient of the square of the semi-major axis divided by the abscissa, and the abscissa; the origin of the co-ordinates being at C, the centre.
456. - Ellipse: Relation of Tangent with the Foci. - Let the two lines from the foci to P (Fig. 302), any point in the ellipse, be extended beyond P. With the radius PF' describe from P the arc F' G, and bisect it in H. Then the line P T, drawn through H, will be a tangent to the ellipse at P.
This has been shown by writers on Conic Sections. The construction here shown affords a ready method of drawing a tangent. And from the principle here given we learn that a tangent makes equal angles with the lines from the tangential point to the two foci.
For, because GH = HF', we have the angle F' PH = HPG. The angles HPG and KPF are opposite, and hence (Art. 344) are equal; and since the two triangles F'PHandKPFare each equal to HPG, therefore F'PH and KPF are equal to each other. Or: A tangent to an ellipse makes equal angles with the two lines drawn from the point of tangency to the two foci.
Experience shows that light shining from one focus is reflected from the ellipse into the other focus. It is for this reason that the two points Fand F' are called foci, the plural of focus, a fireplace.
457. - Ellipse: Relation of Axes to Conjugate Diameters. - Parallel with K T (Fig. 302) let DE be drawn through
C, the centre, and L Q through J, one end of the diameter from the point P. Parallel with this diameter PJ draw L K and Q R through the extremities of the diameter D E. Then D E is a diameter conjugate to the diameter PJ, and K R, R Q, QL, and L K are tangents at the extremities of these conjugate diameters.
Now it is shown by writers on Conic Sections (Fig. 302) that -
a2 + b2 = a'2 + b'2;
Or: The sum of the squares of the two axes equals the sum of the squares of any two conjugate diameters.
From this it is also shown that the area of the parallelogram K C equals the rectangle A C x B C; or, that a parallelogram formed by tangents at the extremities of any two conjugate diameters is equal to the rectangle of the axes.