Having, therefore, the sum and the product of a' and b', we can form the quadratic which determines these quantities. Ex. 1. Find the axes of the ellipse 14x2 - 4xy + 11y2 = 60, and transform the equation to them. The axes are (Art. 155) 4x2 + 6xy - 4y2 = 0, or (2x − y) (x + 2y) = 0. We have a' + b′ = 25; a'b' : is 2x2 + 3y2 = 12. = 150; a' = 10; b′ = 15; and the transformed equation Ex. 2. Transform the hyperbola 11x2 + 84xy - 24y2 = 156 to the axes. a' + b' - 13, a'b' =- 2028; a' = 39; b' = - 52. Transformed equation is 3x2 - 4y2 = 12. Ex. 3. Transform ax2 + 2hxy+by2c to the axes. Ans. (a + b - R) x2 + (a + b + R) y2 = 2c, where R2 = 4h2 + (a - b)2. *158. Having proved that the quantities a+b and ab – h3 remain unaltered when we transform from one rectangular system to another, let us now inquire what these quantities become if we transform to an oblique system. We may retain the old axis of x, and if we take an axis of y inclined to it at an angle w, then (Art. 9) we are to substitute x + y cosa for x, and y sin w We shall then have for y. If, then, we transform the equation from one pair of axes to any other, the quantities a+b-2h cos w ab - h2 sin'w remain unaltered. sin'w We may, by the help of this theorem, transform to the axes an equation given in oblique coordinates, for we can still express the sum and product of the new a and b in terms of the old coefficients. Ex. 1. If cos, transform to the axes 10x2 + 6xy + 5y2 : Ans. 16x+41y2 = 32. Ex. 2. Transform to the axes x2 - 3xy + y2+1=0, where w = = 60°. Ans. 2-15y2 = 3. *159. 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 ax+2hxy + by becomes a'X2+2h'XY+U'Y2. Now we know that the effect of the same substitution will be to make the quantity + 2xy cosw+y become X+2XY cos + Y2, since either is the expression for the square of the distance of any point from the origin. It follows, then, that ax2+2hxy + by2 +λ(x2 + 2xy cose + y2) = a'X2 + 2h' X Y + b′ Y2 + λ (X2 + 2XY cos And if we determine λ 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 (a + λ) (b + λ) = (h + λ cos w)2, or λ must be one of the roots of the equation λ" sin'w + (a+b-2h cos w) λ + ab - h2 = 0. We get a quadratic of like form to determine the value of X, which will make the second side of the equation a perfect square; but since both sides become perfect squares for the same values of A, these two quadratics must be identical. Equating, then, the coefficients of the corresponding terms, we have, as before, a+b-2h cos w ab-h a'b' -h" sin22 sin'w a+b'-2h' cos = sin Ω Ex. 1. The sum of the squares of the reciprocals of two semi-diameters at right angles to each other is constant. Let their lengths be a and ẞ; then making alternately x = 0, y = 0, in the equation of the curve, we have aa2c, bc, and the theorem just stated is only the geometrical interpretation of the fact that a + b is constant. Ex. 2. The area of the triangle formed by joining the extremities of two conjugate semi-diameters is constant. Ex. 3. The sum of the squares of two conjugate semi-diameters is constant. THE EQUATION REFERRED TO THE AXES. 160. We saw that the equation referred to the axes was of the form Ax2 + By2=C, B being positive in the case of the ellipse, and negative in that of the hyperbola (Art. 138, Ex. 3). We have replaced the small letters by capitals, because we are about to use the letters a and b with a different meaning. 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= y=b, then making y=0 and x=a in the equation of the curve, stituting these values, the equation of the ellipse may be written x2 y* Since we may choose whichever axis we please for the axis of x, we shall suppose that we have chosen the axes so that a may 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 y3, 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 y'--', is imaginary; the axis of y, therefore, does not meet the curve in real points. Since we have chosen for our axis of x the axis which meets the curve in real points, we are not in this case entitled to assume that a is greater than b. 161. To find the polar equation of the ellipse, the centre being the pole. Write p cose for x, and p sine for y in the preceding equation, and we get 1 2 cos 0 = a2 sin 0 b2 an equation which we may write in any of the equivalent forms, 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. 162. To investigate the figure of the ellipse. The least value that b2+(a - b) sin'e, the denominator in the value of p', can have, is when 0=0; therefore the greatest value of p is the intercept on the axis of x, and is = a. 0 = Again, the greatest value of b2 + (a* — b2) sin20 is when sin=1, or = 90°; hence, the least value of p is the intercept on the axis of y, and is b. The greatest line, therefore, that can be drawn through the centre is the axis of x, and the least line the axis of y. From this property these lines are called the axis major and the axis minor of the curve. It is plain that the smaller 0 is, the greater p will be; hence, the nearer any diameter is to the axis major, the greater it will be. The form of the curve will, therefore, be that here represented. =- α. B P A M N B We obtain the same value of p whether we suppose 0 = a, or 0 = 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 semi-diameters 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. Y 163. 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==√(a" - 2"). Now, if we describe a concentric circle with the radius a its equation will be 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. A D B M N B D' Hence the circle described on the axis minor lies wholly within the curve. The equation of the circle is the particular form which the equation of the ellipse assumes when we suppose b=a. = 164. To find the polar equation of the hyperbola. Transforming to polar coordinates, as in Art. 161, we get a2b = a2b2 = a2b2 bcos 0-a sin"0 b2 − (a* + b2) sin2 (a2 + b2) cos2 0 − a2 · Since formulæ concerning the ellipse are altered to the corresponding formulæ for the hyperbola by changing the sign of b2, 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 a2 hyperbola. Dividing then by a2 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. p2 = b9 |