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15. We next proceed to consider the results to be deduced from the theory of reciprocal polars, when the auxiliary conic is a circle. It is here that the utility of the theory is most apparent, as we are thus enabled to transform theorems relating to the magnitudes of lines and angles.
We know that, if PQ be the polar of a point T with respect to a circle, of which the centre is S and radius k, then ST will be perpendicular to PQ. Let ST cut PQ in V. Then
ST. SV=k. Hence the pole of any line is at a distance from the centre of the auxiliary circle inversely proportional to the distance of the line. And conversely, the polar of any point is at a distance from the centre of the auxiliary circle, inversely proportional to the distance of the point itself.
16. If TX, TY be any two indefinite straight lines, P, Q their poles, then, since SP is perpendicular to TX, SQ to TY, it follows that the angle PSQ is equal to the angle XTY or its supplement, as the case may be. Hence, the angle included between any two straight lines is equal to the angle subtended at the centre of the auxiliary circle by the straight line joining their poles, or to its supplement.
17. From what has been said in Art. 15, and the earlier articles of this chapter, it will appear that to find the polar reciprocal of a given curve with respect to a circle, we may proceed by either of the following two methods.
First. Draw a tangent to the curve, and from S, the centre of the auxiliary circle, draw SY perpendicular to the tangent, and on SY, produced if necessary, take a point Q, such that SQ.SY=k. The locus of Q will be the required polar reciprocal.
Secondly. Take a point Pon the curve, and join SP; on SP, produced if necessary, take a point Z, such that
SP. SZ=k. Through Z draw a straight line perpendicular to SP. The envelope of this line will be the required polar reciprocal.
18. To find the polar reciprocal of one circle with re
spect to another.
From what has already been shewn, we know that this will be a conic; we have now to investigate its form and position.
Let S be the centre of the auxiliary circle, S its centre, k its radius, MPM' the circle to be reciprocated, O its centre, MM' its diameter passing through S, p its radius.
Through S draw any straight line cutting MPM' in P and Q.
On SPQ, produced if necessary, take two points Y and Z, such that
SP. SY = SQ. SZ=k.
The straight lines drawn through Y and Z perpendicular to SP will be tangents to the reciprocal conic.
K* Now SY. Sz=
SP.SQp* - %! which is constant. Hence, the reciprocal is a conic of such a nature that the rectangle under the distances from S of any two parallel tangents is constant. It is therefore a conic, of
2k2 which S is a focus, and of which the axis-minor is
(p2 – c2) It will be an ellipse, parabola, or hyperbola, according as p is greater than, equal to, or less than c, that is, according as the centre of the auxiliary circle lies within, without, or upon, that to be reciprocated. This agrees with what has been already shewn, Art. 9.
Let 2a, 2b, be the axes of the conic, 21 its latus-rectum, e its eccentricity.
To determine their magnitudes, we proceed as follows. The axis-major will be in the direction So. Let A, A' be its extremities.
SM + SM' 2p
ka Hence, l= or the latus-rectum is inversely propor
р tional to the radius of the circle.
Thus the eccentricity varies directly as the distance between the centres of the circles, and inversely as the radius of that to be reciprocated. If d be the distance from S of the corresponding directrix,
1 R* 2 p ka d
р or, the directrix is the polar of the centre of the circle MPM'.
If S lie without the circle MPM', the angle between the asymptotes of the reciprocal hyperbola will be the supplement of that between the tangents drawn from S to the circle MPM'. (See Art. 9 of this Chapter.)
19. It will be observed that the magnitude of the radius of the auxiliary circle affects the absolute, but not the relative, magnitudes, or positions of the various lines in the reciprocal figure. As our theorems are, for the most part, independent of absolute magnitude, we may generally drop all consideration of the radius of the auxiliary circle, and consider its centre only. We may then speak of reciprocating “ with respect to s," instead of "with respect to a circle of which S is the centre."
20. We have now the means of obtaining, from any property of a circle, a focal property of a conic section. Take, for example, Euc. III. 21. This may be expressed as follows: “If three points be taken on the circumference of circle, two fixed and the third moveable, the straight lines joining the moveable point with the two fixed points, make a constant angle with one another.” This will be reciprocated into “If three tangents be drawn to a conic section, two fixed and the third moveable, the portion of the moveable tangent intercepted between the two fixed ones, subtends a constant angle at the focus.” This angle will be found, by reciprocating Euc. III. 20, to be one-half of the angle subtended at the focus by the portion of the corresponding directrix intercepted between the two fixed tangents.
Again, it is easy to see that "if a circle be described touching two concentric circles, its radius will be equal to half the sum, or half the difference, of the radii of the given circles, and the locus of its centre will be a circle, concentric with the other two, and of which the radius is half the difference, or half the sum, of the radii of the two given circles.”
Hence we deduce the following theorem. “If two conics have a common focus and directrix, and their latera-recta be 21, 21, and another conic, having the same focus, be described
4ll' so as to touch both of them, its latus-rectum will be and the envelope of its directrix will be a conic, having the same focus and directrix as the given conics, and of which
411' the latus-rectum is
Again, take the ordinary definition of an ellipse, that it is the locus of a point, the sum of the distances of which from two fixed points is constant. This is equivalent to “the sum of the distances from either focus, of the points of contact of two parallel tangents, is constant.
The reciprocal theorem will be, "If a system of chords be drawn to a circle, passing through a given point, and, at the extremities of any chord, a pair of tangents be drawn to the circle, the sum of the reciprocals of the distances of these tangents from the fixed point is constant."
The known property of a circle that “two tangents make equal angles with their chord of contact” will be found, when transformed by the method now explained, to be equivalent to the theorem that “if two tangents be drawn to a conic from an external point, the portions of these tangents, intercepted between that point and their points of contact, subtend equal angles at the focus."
21. Two points, on a curve and its reciprocal, are said to correspond to one another when the tangent at either point is the polar of the other point. Two tangents are said to