By means of the relation b2-4ac=0, it is easily shewn that the second form of the conditions coincides with the first when a and c are both different from zero. When a = 0 the first is the necessary form of the conditions, but we see that the second form will then also hold. When c=0 the second is the necessary form, though the first will then also hold. Hence we shall include every case by stating that both forms of the conditions must hold. Similarly the conditions under which the locus will consist of one straight line, or will be impossible, may be investigated. 280. We will recapitulate the results of the present chapter with respect to the locus of the equation ax2 + bxy + cy2+ dx+ey +ƒ=0. I. If b2 - 4ac be negative, the locus is an ellipse admitting of the following varieties: (1) c = a, and b 2a = locus a circle (Art. 104). cosine of the angle between the axes; (2) (e2 - 4cf") (b2 — 4ac) — (be — 2cd) positive; locus impossible. (3) (e2 — 4cf') (b2 — 4ac) — (be — 2cd)2= 0; locus a point. II. If b2-4ac be positive, the locus is an hyperbola, except when (b2 — 4ac) ƒ+ ae2 + cd2 — bde = 0, and then it consists of two intersecting straight lines. III. If b2-4ac = 0, the locus is a parabola, except when be-2cd=0, and bd-2ae=0; and then it consists of two parallel straight lines, or of one straight line, or is impossible, according as e-4cf and d-4af are positive, zero, or negative. EXAMPLES. 1. Find the centre of the curve x2 — 4xy +4y2 — 2ax + 4ay = 0. 2. Find the centre of the ellipse by (1-2)+cx (1-7)=xy. 3. What is represented by ax2+2bxy+cy2 = 1, when b2 = ac? 4. Find the locus of the centre of a circle inscribed in a sector of a given circle, one of the bounding radii of the sector remaining fixed. 5. In the side AB of a triangle ABC, any point P is taken, and PQ is drawn perpendicular to AC; find the locus of the point of intersection of the straight lines BQ and CP. 6. DE is any chord parallel to the major axis AA' of an ellipse whose centre is ; and AD and CE intersect in P; shew that the locus of P is an hyperbola, and find the direction of its asymptotes. 7. Tangents to two concentric ellipses, the directions of whose axes coincide, are drawn from a point P, and the chords of contact intersect in Q; if the point P always lies on a straight line, shew that the locus of Q will be a rectangular hyperbola. 8. What form does the result in the preceding example take when two of the axes whose directions are coincident are equal? 9. Prove that an hyperbola may be described by the intersection of two straight lines which move parallel to themselves while the product of their distances from a fixed point remains constant. 10. Two lines are drawn from the focus of an ellipse including a constant angle; tangents are drawn to the ellipse at the points where the lines meet the ellipse; find the locus of the intersection of the tangents. 11. Find the latus rectum of the parabola (y — x)2 = ax. 12. Shew that the product of the semi-axes of the ellipse - 4xy+5x2= 2 is 2. 13. Find the angle between the asymptotes of the hyperbola xy=bx + c. 14. Find the equation to a parabola which touches the axis of x at a distance a, and cuts the axis of y at distances B, B' from the origin. 15. If two points be taken in each of two rectangular axes, so as to satisfy the condition that a rectangular hyperbola may pass through all the four, shew that the position of the hyperbola is indeterminate, and that its centre describes a circle which passes through the origin and bisects all the lines which join the points two and two. 16. Two lines of given lengths coincide with and move along two fixed axes in such a manner that a circle may always be drawn through their extremities; find the locus of the centre of the circle, and shew that it is an equilateral hyperbola. 17. A variable ellipse always touches a given ellipse, and has a common focus with it; find the locus of its other focus, (1) when the major axis is given, (2) when the minor axis is given. 18. Draw the curve y2 — 5xy + 6x2 - 14x + 5y + 4 = 0. 19. Draw the curve 20. Find the nature and position of the curve 21. The equation to a conic section being ax2+2bxy + cy2 = 1, shew that the equation to its axes is xy (a−c) = b (x2 — y3). 22. The locus of the vertices of all similar triangles whose bases are parallel chords of a parabola will in general be another parabola; but if any one of the triangles touch the parabola with its sides, the locus becomes a straight line. 23. A series of circles pass through a given point_O, have their centres in a line ÒA, and meet another line BC. Let M be the point in which one of the circles meets the line OA again, and let N be either of the points in which this circle meets BC. From M and N lines are drawn parallel to BC and OA respectively, intersecting in P; shew that the locus of P is an hyperbola which becomes a parabola when the two lines are at right angles. 24. The chord of contact of two tangents to a parabola subtends an angle ẞ at the vertex; shew that the locus of their point of intersection is an hyperbola whose asymptotes are inclined to the axis of the parabola at an angle & such that tan = 1 tan B. 25. Determine the locus of the middle points of the chords of the curve ax2+2bxy + cy2 + 2ex + 2fy + g=0, which are parallel to the line x sin 0 — y cos 0=0; and hence find the position of the principal axes of the curve. 26. Shew that the equation (x2 — a3)2 + (y2 — a3)2 = a* represents two ellipses. CHAPTER XIV. MISCELLANEOUS PROPOSITIONS. 281. WE shall give in this chapter some miscellaneous propositions for the most part applicable to all the conic sections. To find the equation to a conic section, the origin and axes being unrestricted in position. Let a, b be the co-ordinates of the focus; and let the equation to the directrix be Ax+ By + C = 0. The distance of any point (x, y) from the focus is {(x − a)2 + (y — b)2}3, and the distance of the same point from the directrix is Ax+ By + C Let e be the excentricity of the conic section; then if (x, y) be a point on the curve, we have, by definition, We see from (1) that the distance of any point on a conit section from the focus can be expressed in terms of the first |