Then, let CM = x, MP = y, SP=r, HP = r', SH = 2c. If, therefore, we eliminate r and r' between these three equations, we shall obtain a relation between x and y, which will be the required equation to the Sincer and r' or 2a is always → SH or 2c, .. a is always c, and.. the quantity ac2 is essentially positive. which is the most simple form of the equation to the ellipse. Solving the equation for y and x in succession, we obtain the curve is situated symmetrically with regard both to the axes CX and CY. For, taking the first of the two equations, we perceive that for each assumed value of x, we shall obtain two equal values of y with opposite signs, which shows that the curve is situated symmetrically with regard to CX. And in the same manner, taking the second equation, we perceive that for each assumed value of y, we shall obtain two equal values of x with opposite signs. The distance CS = CH is called the Eccentricity of the ellipse, and the ratio of c to a is usually denoted by the symbol e. It is necessary to observe these equations, since the quantity e is very frequently introduced in calculations where the equation to the ellipse is employed. PROB. XII. To find the equation to the ellipse, referred to the vertex as origin. a In order to transport the origin from C to a, since the new origin a is situated in the old axis of x's at a distance = a, we have only to substitute * x for x in equation (A), which then becomes To find the equation to the Hyperbola. DEFINITION. The Hyperbola is the locus of a point, the difference of whose distance from two given fixed points is equal to a constant quantity. Let S and H be the two given fixed points. Let P be any point in the curve, join S, P; H, P ; Draw CY perpendicular to CX, and let C be the origin, and CX, CY, the axes of co-ordinates. Let the constant quantity to which the difference of SP and HP is always equal be 2a. Let CM = x, MP = y, SP=r, HP=r', SH = 2c Y C A S (1) (2) (3). If we eliminate r and r' between these three equations, we shall arrive at an equation between x and y, which will be the equation to the curve. *See chapter on the "Transformation of Co-ordinates," Then, let CM = x, MP = y, SP=r, HP = r', SH = 2c. r+r' = 2α If, therefore, we eliminate r and r' between these three equations, we shall obtain a relation between x and y, which will be the required equation to the Sincer and r' or 2a is always → SH or 2c, .. a is always c, and .. the quantity a2-c2 is essentially positive. which is the most simple form of the equation to the ellipse. Solving the equation for y and x in succession, we obtain b When y = ± Va2 (B) Hence it appears that the curve cuts the axis of r's at the points A, a, where CA Ca =a, and cuts the axis of y's at the points B, b, where CB Cb = √ a2 — c2 = b. the curve is situated symmetrically with regard both to the axes CX and CY. For, taking the first of the two equations, we perceive that for each assumed value of x, we shall obtain two equal values of y with opposite signs, which shows that the curve is situated symmetrically with regard to CX. And in the same manner, taking the second equation, we perceive that for each assumed value of y, we shall obtain two equal values of x with opposite signs. The distance CS = CH is called the Eccentricity of the ellipse, and the ratio of c to a is usually denoted by the symbol e. It is necessary to observe these equations, since the quantity e is very frequently introduced in calculations where the equation to the ellipse is employed. PROB. XII. To find the equation to the ellipse, referred to the vertex as origin. * a In order to transport the origin from C to a, since the new origin a is situated in the old axis of a's at a distance = - a, we have only to substitute х for x in equation (A), which then becomes DEFINITION.-The Hyperbola is the locus of a point, the difference of whose distance from two given fixed points is equal to a constant quantity. Let S and H be the two given fixed points. Join S, H; and bisect SH in C. Let P be any point in the curve, join S, P ; H, P ; Draw CY perpendicular to CX, and let C be the origin, and CX, CY, the axes of co-ordinates. Let the constant quantity to which the difference of SP and HP is always equal be 2a. Let CM = x, MP = y, SP=r, HP=r', SH = 2c Then y2+ (x − c )2 = 72 y2+(x+c)2 r_r2a..... ........ A S M X (1) (3). If we eliminate r and r' between these three equations, we shall arrive at an equation between x and y, which will be the equation to the curve. * See chapter on the "Transformation of Co-ordinates," or, y2+ x2 + c2 = a2 + @2 a2y2 + (a2 — c2) x2 = a2 (a3 — c"). Now 2a must always be 2c, and therefore a always c. Hence a2 c2 is essentially negative. c2 - -- which is the most simple form of the equation to the hyperbola. Let b y = ± a √(x* — a3) α x = ± 7 √ y2 + b2 y = 0.: x = + a. (D) (E) From which it appears that the curve cuts the axis of x, at the points A, a; where CA Ca= a. Let = x=0..y = ±b√= 1, an impossible result. From which it appears that the curve does not meet the axis of y's. We may, however, take two points B, b, in this axis on different sides of C making CB = Cb = √c2 In order to fix the position of these points, from the point A as centre with radius equal to CS CH, describe a circle which will cut the line YY' in the two points required; for we have CB= √ CS2 — CA2 = √/ c2 — a2. Hence it appears, that the quantities a and b are the semi-major and semiminor axes of the hyperbola, and hence the equation (D) is called the equation to the Hyperbola referred to its axes. Resuming the equation (E), we perceive that if x be▲a, the values of y are impossible; and hence we conclude, that there is no point in the curve situated between A and a. When a isa, then as x increases y increases also; and for each value of r, here will be two equal values of y with opposite signs. Hence, it is evident from the equation, that the hyperbola consists of two opposite branches, one extending indefinitely to the right of A, and the other indefinitely to the left of a, and both symmetrically situated with regard to XX. |