Ex. 3. Transform Ax2 + Bxy + Cy2 = F to the axes. Ans. (ACB cos w - R) x2 + (A + C B cos w + R) y2 = 2F sin3w, where R2 = {B - (A + C) cos w } 2 + (A − C)2 sin2w. - *162. We add the demonstration of the theorems of the last two articles given by Professor Boole (Cambridge Math. Jour., iii. 1, 106, and New Series, vi. 87). Let us suppose that we are transforming an equation from axes inclined at an angle w, to any other axes inclined at an angle ; and that, on making the substitutions of Art. 9, the quantity Ax2 + Bay + Cy2 becomes A'X2+ B'XY+ CY2. Now we know that the effect of the same substitution will be to make the quantity x2 + 2xy cos w + y2 become X2 + 2XY cosQ + Y2, since either is the expression for the square of the distance of any point from the origin. It follows, then, that = Ax2 + Bxy + Cy2 + h (x2 + 2xy cos w + y2 ) · A'X2 + B'XY + C'Y2 + h (X2 + 2XY cos Q + Y2). And if we determine h so that the first side of the equation may be a perfect square, the second must be a perfect square also. But the condition that the first side may be a perfect square is (B+ 2h cos w)2 = 4 (A + h) (C + h), or h must be one of the roots of the equation 4h2 sin2 + 4 (A + C − B cos w) h + 4AC – B2 = 0. We get a quadratic of like form to determine the value of h, which will make the second side of the equation a perfect square; but since both sides become perfect squares for the same values of h, these two quadratics must be identical. Equating, then, the coefficients of the corresponding terms, we have, as before, A+C-B cos w A'+C'B' cos Q B2-4AC B'2-4A'C' sin2w sin2Q ; sin2w sin Q Ex. 1. The sum of the squares of the reciprocals of two semidiameters at right angles to each other is constant. Let their lengths be a and b; then making alternately x = = 0, y = 0, in the equation of the curve, we have Aa2 F, Cb2 F, and the theorem just stated is only the geometrical interpretation of the fact that A+ C is constant. Ex. 2. The area of the triangle formed by joining the extremities of two conjugate semidiameters is constant. The equation referred to two conjugate diameters is is constant, we have a'b' sino constant. 22 y2 1, and since 4AC-B2 sin2w Ex. 3. The sum of the squares of two conjugate semidiameters is constant. A+CB cos w is constant, 1 1 sin2w 7'2 is constant; and since a'b' sin w Since is constant, so must a2+ b2. THE EQUATION REFERRED TO THE AXES. 163. We saw that the equation referred to the axes was of the form Ax2 + Cy2 = F, C being positive in the case of the ellipse, and negative in that of the hyperbola (Art. 136, 11.) The equation of the ellipse may be written in the following more convenient form : Let the intercepts made by the ellipse on the axes be x = a, y = b, then a is found by making y = 0 and x = a in the equation F b2 of the curve, or Aa2 = F, and A F = In like manner, C a2 Substituting these values, the equation of the ellipse may be Since we may choose whichever axis we please for the axis of have chosen the axes so that a may x, we shall that we suppose be greater than b. The equation of the hyperbola, which, we saw, only differs from that of the ellipse in the sign of the coefficient of y2, may be written in the corresponding form, The intercept on the axis of x is evidently = ± a, but that on the axis of y, being found from the equation y2= b2 is imaginary; the axis of y, therefore, does not meet the curve in real points. Since we have chosen for our axis of a the axis which meets the curve in real points, we are not in this case entitled to assume that a is greater than b. 164. To find the polar equation of the ellipse, the centre being the pole. Write p cose for a, and p sin 0 for y, in the preceding equation, and we get 1 cos20 sin20 an equation which we may write in any of the equivalent forms, p2 a2 b2 = a2 b2 a2 b2 a2 sin20+ b2 cos20 b2 + (a2 – b2) sin30 ̄ ̄ a2 - (a2-b2) cos 0 It is customary to use the following abbreviations, and the quantity e is called the eccentricity of the curve. Dividing by a the numerator and denominator of the fraction last found, we obtain the form most commonly used, viz., b2 1 - e2 cos20 p2 165. To investigate the figure of the ellipse. The least value that b2+ (a2b2) sin20 can have, is when 00; therefore, since = the greatest value of p is the intercept on the axis of x, and is = a. Again, the greatest value of b2 + (a2 – b2) sin20, is, when sin = 1, or 0 = 90°; hence the least value of p is the intercept on the axis of y, and is = b. The greatest line, therefore, that can 0 == any diameter is to the axis major, the greater it will be. The form of the curve will, therefore, be that here represented. U 0 = We obtain the same value of p whether we suppose = a, or α. Hence, Two diameters which make equal angles with the axis will be equal. And it is easy to show that the converse of this theorem is also true. This property enables us, being given the centre of a conic, to determine its axes geometrically. For, describe any concentric circle intersecting the conic, then the semidiameters drawn to the points of intersection will be equal; and by the theorem just proved, the axes of the conic will be the lines internally and externally bisecting the angle between them. 166. The equation of the ellipse can be put into another form, which will make the figure of the curve still more apparent. If we solve for y we get b y = = - √ (a2 - x2). a Now, if we describe a concentric circle with the radius a, its equation will be y = √ (a2 − x2). - Hence we derive the following construction : "Describe a circle on the axis major, and take on each ordinate LQ a point P, such that LP may be to LQ in the constant ratio b: a, then the locus of P will be the required ellipse." Hence the circle described on the axis major lies wholly without the curve. We might, in like manner, construct the ellipse, by describing a circle on the axis minor, and increasing each ordinate in the constant ratio a:b. Hence the circle described on the axis minor lies wholly within the curve. A Ꭰ B M N B D' T A The equation of the circle is the particular form which the equation of the ellipse assumes when we suppose b = a. 167. To find the polar equation of the hyperbola. Transforming to polar co-ordinates, as in Art. 164, we get Since formulæ concerning the ellipse are altered to the corres a2 ponding formula for the hyperbola by changing the sign of b', we must, in this case, use the abbreviation c2 for a2 + b2, and a2 + b2 e2 for the quantity e being called the eccentricity of the hyperbola. Dividing then by a the numerator and denominator of the last found fraction, we obtain the polar equation of the hyperbola, which only differs from that of the ellipse in the sign of b2, viz., b2 p2 e2 cos20-1' 168. To investigate the figure of the hyperbola. The terms axis major and axis minor not being applicable to the hyperbola (Art. 163), we shall call the axis of x the transverse axis, and the axis of y the conjugate axis. Now b2 - (a2 + b) sin2 0, the denominator in the value of p2, will plainly be greatest when 0 = 0, therefore, in the same case, p will be least; or the transverse axis is the shortest line which can be drawn from the centre to the curve. As increases, p continually increases, until ρ when the denominator of the value of p becomes = 0, and p becomes infinite. After this value of 0, p2 becomes negative, and the diameters cease to meet the curve in real points until when p again becomes infinite. It then decreases regularly as increases, until ◊ becomes = 180°, when it again receives its minimum value =α. The form of the hyperbola, therefore, is that represented by the dark curve on the figure. y2 = b2 to determine its point of intersection with the curve. |