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 = a, y=b, then making y=0 and x = a in the equation of the curve, we have Aa=C, and A=G. In like manner B= C Sub stituting these values, the equation of the ellipse may be written 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: х 1. 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 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 cos20 sin' Ꮎ 2 + a2 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'0, the denominator in the value of p2, can have, is when 0=0; therefore the greatest value of p is the intercept on the axis of x, and is = a. = Again, the greatest value of b2 + (a* — 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 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. the It is plain that the smaller is, the greater p will be; hence, 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 AM N We obtain the same value of p whether we suppose θα =a, or 0 Hence, Two diameters which make equal angles with the axis will be equal. that the converse of this theorem is also true. B And it is easy to show 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. 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 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 A 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 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 b', viz. 165. To investigate the figure of the hyperbola. The terms axis major and axis minor not being applicable to the hyperbola (Art. 160), we shall call the axis of x the transverse axis, and the axis of y the conjugate axis. Now b2 (a+b2) sin*0, the denominator in the value of p2, will plainly be greatest when 0=0, therefore, in the same case, ρ 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 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 again when p again becomes infinite. It then decreases regularly as increases, until becomes = 180°, when it again receives its minimum value = a. The form of the hyperbola, therefore, is that represented by the dark curve on the figure, next article. = K 166. We found that the axis of y does not meet the hyperbola in real points, since we obtained the equation y2-b2 to determine its point of intersection with the curve. We shall, however, still mark off on the axis of y portions CB, CB'=±b, and we shall find that the length CB has an important connexion with the B IC A B curve, and may be conveniently called an axis of the curve. In like manner, if we obtained an equation to determine the length of any other diameter, of the form p2=- R2, although this diameter cannot meet the curve, yet if we measure on it from the centre lengths + R, these lines may be conveniently spoken of as diameters of the hyperbola. = The locus of the extremities of these diameters which do not meet the curve is, by changing the sign of p" in the equation of the curve, at once found to be This is the equation of a hyperbola having the axis of y for the axis meeting it in real points, and the axis of x for the axis meeting it in imaginary points. It is represented by the dotted curve on the figure, and is called the hyperbola conjugate to the given hyperbola. 167. We proved (Art. 165) that the diameters answering to b tan meet the curve at infinity; they are, therefore, the 0 α same as the lines called, in Art. 154, the asymptotes of the curve. They are the lines CK, CL on the figure, and evidently separate those diameters which meet the curve in real points from those which meet it in imaginary points. It is evident also that two conjugate hyperbolæ have the same asymptotes. b α The expression tan 0 = ± =+ enables us, being given the axes in magnitude and position, to find the asymptotes, for if we form a rectangle by drawing parallels to the axes through B and A, then the asymptote CK must be the diagonal of this rectangle, But, since the asymptotes make equal angles with the axis of x, the angle which they make with each other must be = 20. Hence, being given the eccentricity of a hyperbola, we are given the angle between the asymptotes, which is double the angle whose secant is the eccentricity. Ex. To find the eccentricity of a conic given by the general equation. We can (Art. 74) write down the tangent of the angle between the lines denoted by ax2 + 2hxy + by2 = 0, and thence form the expression for the secant of its half; or we may proceed by the help of Art. 157, Ex. 3. |