131. The next articles will be occupied with a discussion of the different forms assumed by the quadratic just found for р, according to the different values we may give the ratio m:n. The reader will better understand the method we pursue if he bear in mind the following elementary principles. Suppose that we have to discuss any quadratic, ap2 + bp + c = 0, its solution may be written in either of the following equivalent forms, the latter being the form in which the solution would have presented itself had we divided the given equation by p2, and solved it for the reciprocal of p. 1. If we have c = 0, the quadratic is divisible by p, and one of its roots is p = 0, the other being b If we had not only c = 0, a but also b = 0, then the quadratic would be divisible by p2, and both its roots would 0. 11. If we have a = 0, then one of the roots of the equation is For if we had written the equation p = ∞. same thing may be seen by making a = 0 in the general form of the solution. If not only a = 0, but also b = 0, both the roots =∞. III. If b= 0, the roots of the quadratic are equal with opposite signs. Iv. If we have b2 = 4ac, the two roots are equal, and we may roots of the quadratic are real; ifb be less than 4ac, the roots are imaginary. R 132. Let us now apply these principles to the equation which determines the points where the line (my = nx) meets the curve, viz. (Am + Bmn + Cr)p* + (Dm + En)p + F = 0. = 1. Let F = 0. In this case one of the values of o is ρ 0, or the origin is one of the points where the line meets the curve (see also Art. 79). The other value is = If, however, we have not only F = 0, but also the line be drawn in such a direction that Dm + En = 0, then the second value of p is also 0: the line (mynx) meets the curve in two coincident points at the origin, or, in other words, is a tangent at the origin. Multiplying by p the equation Dm + En = 0, and remembering that mp = x, np = y, we find the equation of the tangent at the origin, viz. Dx + Ey = 0. ρ transform the equation to parallel axes through this point, and find the tangent at it. Ans. 9x 5y = 0 referred to the new axes, or 9 (x − 1) — 5 (y − 1) = 0 referred to the old. 133. To find the equation of the tangent at any point x'y' on the curve. Transform to parallel axes through x'y', and (Art. 129) F' will vanish, since x'y' is on the curve. The equation of the tangent will then be D'x + Ey = 0 referred to the new axes, or D'(x-x') + E' (y-y') = 0 referred to the old. "Write for D' and E' the values found in Art. 129, and the equation of the tangent is (2Ax' + By' + D) (x − x') + (Bx' + 2Cy' + E) (y − y) which may be written in a simpler form by adding to both sides the identity 2 Ax22 + 2Bx'y' + 2Cy2 + 2Dx′ + 2Ey' + 2F when the equation of the tangent becomes 0, = 0, (2Ax+ By + D) x + (Bx' + 2Cy' + E) y + Dx' + Ey' + 2F = 0. * This equation might also have been found by the method pursued in Art. 84. Ex. Find the tangent at (2, 1) to 3x2 + 4xy + 5y2 — 7x - 8y - 3 = 0. Ans. 9x+10y= 28. 134. 11. Let us next consider the case in which one value of p may become infinite. We have seen (Art. 131) that this will be the case when the coefficient of p2 vanishes in the quadratic which determines p; or, in other words, when (my = Am2 +Bmn + Cn2 = 0. If then m:n be taken so as to satisfy this relation, the line nx) will meet the curve in one infinitely distant point: the other value of p will in general remain finite, and will F Dm + En Since two values of m: n can in general be found, which will render Am2 + Bmn + Cn2 = 0, there can be drawn through the origin two real, coincident, or imaginary lines, which will meet the curve at an infinite distance, and each of these lines will only meet the curve in one other point. If we multiply by p2 the equation Am2 + Bmn + Cn2 0, and substitute for and mp ηρ their values x and y, we obtain for the equation of these two lines, = Ax2 + Bxy + Cy2 = 0. 2 We may prove, by the transformation of co-ordinates, as in Art. 130, that there are two directions in which lines can be drawn through any point to meet the curve at infinity; and, since it was proved, in Art. 129, that the coefficients A, B, C were unaltered by transformation, we obtain for every point the very same quadratic, Am2 + Bmn + Cn2 = 0, to determine those directions. Hence, if through any point two real lines can be drawn to meet the curve at infinity, parallel lines through any other point will meet the curve at infinity.* * This, indeed, is evident geometrically, since parallel lines may be considered as passing through the same point at infinity. 135. The most important question we can ask, concerning the form of the curve represented by any equation, is, whether it be limited in every direction, or whether it extend in any direction to infinity. We have seen, in the case of the circle, that an equation of the second degree may represent a limited curve, while the case where it represents right lines shows us that it may also represent loci extending to infinity. It is necessary, therefore, to find a test whereby we may distinguish which class of locus is represented by any particular equation of the second degree. With such a test we are at once furnished by the last article. For if the curve be limited in every direction, no radius vector drawn from the origin to the curve can have an infinite value; but we found in the last Article, that, in order that the radius vector should become infinite, we must have Am2 + Bmn + Cn2 = 0. (1.) If now we suppose B2 - 4AC to be negative, the roots of this equation will be imaginary, and no real value of m: n can be found which will render Am2 + Bmn + Cn2 = 0. In this case, therefore, no real line can be drawn to meet the curve at infinity, 10. and the curve will be limited in every direction. We shall show, in the next chapter, that its form is that represented in the figure. A curve of this class is called an Ellipse. (2.) If B2 – 4AC be positive, the roots of the equation Am2 + Bmn + Cn2 = 0 will be real; consequently, there are two real values of m:n which will render infinite the radius vector to one of the points where the line (mynx) meets the curve. Hence, two real lines (Ax2 + Bxy + Cy2= 0) Y X can, in this case, be drawn through the origin to meet the curve at infinity. A curve of this class is called an Hyperbola, and we shall show, in the next chapter, that its form is that represented in the figure. (3.) If B2-4AC = 0, the roots of the equation Am2 + Bmn + Cn2 = 0 will then be equal, and, therefore, the two directions in which a right line can be drawn to meet the curve at infinity will in this case coincide. A curve of this class is called a Parabola, and we shall (Chap. x11.) show that its form is that here represented. X 136. In applying to Examples the principles just laid down, the following are some of the particular cases which most frequently present themselves :— (1.) The circle is a particular form of the ellipse, for, since in the most general form of the equation of the circle C = A, B = 2A cose (Art. 78), we have (2.) If B = 0, the curve will be an ellipse if A and C have the same sign; but an hyperbola if they have different signs. (3.) If either A or C = 0, and B not = 0, the quantity B2- 4AC will reduce to B', which being essentially positive, the curve is an hyperbola. = In the case where A O the axis of x is itself one of the lines which meet the curve at infinity; and where C = 0, the axis of y; these lines being in general given by the equation then B2 - 4AC = 0, and the curve is a parabola. (5.) In general the curve will be a parabola, if the three first terms form a perfect square. Ex. Determine the species of each of the following curves: 3x2 + 4xy + 5y2 - 2x - 7y - 4 = 0. Ans. Ellipse. Ans. Hyperbola. Ans. Parabola. y-1=0. |