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By substitution we see that the three points of intersection are on the circle whose equation is

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The circle always passes through the focus of the parabola.

(5) To find the locus of the foot of the perpendicular from the focus of a conic on any tangent.

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Now, if (r, 0) be the co-ordinates of the foot of the perpendicular from

the origin on (i) or on (ii), we have r=; and 0=A;

E

therefore, from (iii),

cos 0-e=cos a, and - sin 0=sin a.

By the elimination of a we get

12 = (1 − e2) r2 + 2el r cos 0.

The locus is therefore a circle, as we have already found [Art. 125, (n)].

EXAMPLES ON CHAPTER VIII.

1. The exterior angle between any two tangents to a parabola is equal to half the difference of the vectorial angles of their points of contact.

2. The locus of the point of intersection of two tangents to a parabola which cut one another at a constant angle is a hyperbola having the same focus and directrix as the original parabola.

3. If PSP' and QSQ' be any two focal chords of a conic 1 1 at right angles to one another, shew that is constant.

+ PS.SP

QS.SQ'

4. If A, B, C be any three points on a parabola, and the tangents at these points form a triangle A'B'C', shew that SA.SB.SC = SA'. SB'. SC', S being the focus of the parabola.

5. If a focal chord of an ellipse make an angle a with the axis, the angle between the tangents at its extremities is

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=

1+ e cos 0, shew that the

6. By means of the equation ellipse might be generated by the motion of a point moving so that the sum of its distances from two fixed points is constant.

7. Find the locus of the pole of a chord which subtends a constant angle (2a) at a focus of a conic, distinguishing the cases for which cos a >=<e.

8. PQ is a chord of a conic which subtends a right angle at a focus. Shew that the locus of the pole of PQ and the locus enveloped by PQ are each conics whose latera recta are to that of the original conic as √2:1 and 1: √2 respectively.

N

9. Given the focus and directrix of a conic, shew that the polar of a given point with respect to it passes through a fixed point.

10. If two conics have a common focus, shew that two of their common chords will pass through the point of intersection of their directrices.

11. Two conics have a common focus and any chord is drawn through the focus meeting the conics in P, P' and Q, Q' respectively. Shew that the tangents at P or P' meet those at Q, Q' in points lying on two straight lines through the intersection of the directrices, these lines being at right angles if the conics have the same eccentricity.

12. Through the focus of a parabola any two chords LSL, MSM' are drawn; the tangent at I meets those at M, M' in the points N, N' and the tangent at L' meets them in K', K. Shew that the lines KN, K'N' are at right angles,

13. Two conics have a common focus about which one is turned; shew that two of their common chords will touch conics having the fixed focus for focus,

14. Shew that the equation of the locus of the point of intersection of two' tangents to

=

1 + e cos 0, which are at right angles to one another, is 2 (e2 – 1) — 2le r cos 0 + 212 = 0.

15. If PSQ, PHR be two chords of an ellipse through the PS PH

foci S, H, then will + be independent of the position SQ HR

= of P.

16. Two conics are described having the same focus, and the distance of this focus from the corresponding directrix of each is the same; if the conics touch one another, prove that twice the sine of half the angle between the transverse axes is equal to the difference of the reciprocals of the eccentricities.

17. A circle of given radius passing through the focus of a given conic intersects it in A, B, C, D; shew that

is constant.

SA.SB.SC.SD

S. C. S.

12

18. A circle passing through the focus of a conic whose latus rectum is 21 meets the conic in four points whose distances 1 1 1 1

from the focus are 1, r

29

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a prove that

+ +

2 = +

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19. A given circle whose centre is on the axis of a parabola passes through the focus S, and is cut in four points A, B, C, D by any conic of given latus rectum having S for focus and a tangent to the parabola for directrix; shew that the sum of the distances SA, SB, SC, SD is constant.

20. Two points P, Q are taken one on each of two conics, which have a common focus and their axes in the same direction, such that PS and QS are at right angles, S being the common focus. Shew that the tangents at P and Q meet on a conic whose eccentricity is equal to the sum of the squares of the eccentricities of the original conics.

21. A series of conics are described with a common latus rectum; prove that the locus of points upon them, at which the perpendicular from the focus on the tangent is equal to the semi-latus rectum, is given by the equation / =-r cos 20.

22. If POP' be a chord of a conic through a fixed point 0, then will tanP'SO tan PSO be constant, S being a focus of the conic.

a

23. Conics are described with equal latera recta and common focus. Also the corresponding directrices envelope a fixed confocal conic. Prove that these conics all touch two fixed conics, the reciprocals of whose latera recta are the sum and difference respectively of those of the variable conic and their fixed confocal and which have the same directrix as the fixed confocal.

CHAPTER IX.

GENERAL EQUATION OF THE SECOND DEGREE.

166. WE have seen in the preceding Chapters that the equation of a conic is always of the second degree: we shall now prove that every equation of the second degree represents a conic, and shew how to determine from any such equation the nature and position of the conic which it represents.

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167. To shew that every curve whose equation is of the second degree is a conic.

We may suppose the axes of co-ordinates to be rectangular; for if the equation be referred to oblique axes, and we change to rectangular axes, the degree of the equation is not altered [Art. 53].

Let then the equation of the curve be

ax2+2hxy+by2+2gx+2fy + c = 0.........(i).

As this is the most general form of the equation of the second degree it will include all possible cases.

We can get rid of the term containing xy by turning

the axes through a certain angle.

For, to turn the axes through an angle substitute for x and y respectively x cos xsin + y cos 0 [Art. 50].

we have to y sin 0, and

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