Y; and in like manner, if HZ be drawn perpendicular to Tt, it may be proved that the same circle will pass through Z also. PROP. VIII. The rectangle contained by perpendiculars from the foci upon the tangent at any point, is equal to the square of the semi-axis, minor. Let Tt be a tangent at any point P; On Aa describe a circle cutting Tt in Y and Z; join S, Y; H, Z. Then, by last Prop., SY, HZ are perpendicular H to Tt. Let HZ meet the circumference in z; Join C,z; C, Y; Since zZY is a right angle, the segment in which it lies is a semicircle, and z, Y, are the extremities T of a diameter; ..zCY is a straight line and a diameter. Hence the triangles CYS, CzH, are in every respect equal; .. SY Hz ... SY. HZ = Hz. HZ = HA.H Geom. Theor. Gi. BC. Prop. 3. PROP. IX. Perpendiculars let fall from the foci upon the tangent at any point, are to cach other as the focal distance of the point of contact. PROP. X. If a tangent be applied at any point, and from the same point an ordinate to the axis be drawn, the semi-axis major is a mean proportional between the distance from the centre, to the intersection of the ordinate with the axis, and the distance from the centre to the intersection of the tangent with the axis. But since PM is drawn from the vertex of triangle HPS perpendicular to HS produced, HM-SM: HP+ SP :: HP-SP: HM + SM or, or, SH : HP + SP :: 2 AC : 2 CM........ Comparing this with the proportion marked (1), we have 2 CT : 2 AC :: 2 AC : 2 CM ..........(2) which is absurd; ..QM is not a tangent at Q; and in the same manner it may be proved that no line but QM' can be a tangent at Q. PROP XII. The square of any semi-ordinate to the axis, is to the rectangle under the abscissæ, as the square of the semi-axis minor, is to the square of the semi-axis major. That is, if P be any point in the curve, PM: AM. Ma :: BC: AC' Describe a circle on Aa, and draw PT a tangent to the hyperbola at P, intersecting the circle in the points Y, Z, and the major axis in T. Draw TQ perpendicular to Aa, meeting the circle in Q; join QM Then QM is a tangent to the circle at Q by Prop. 11, and .. the angle CQM is a right angle. Join S, Y; H, Z; SY and HZ are perpendicular to Tt, Prop. 7. Hence the triangles PMT, SYT, HZT, are similar to each other. MT2 : TQ', Prop. 8, and Geom. Theor. 61. :: AM. Ma: .. PM: AM.Ma :: : AC1 That is, the square of the ordinates to the axis are to each other as the rectangles of their abscissa. Cor. 2. By Prop. PM 2: AM. Ma :: BC2: AC' But AM=CM — CA, Ma — CM + CA, 、 .. PM2: CM2 – CA :: BC2: AC PROP. XIII. The latus rectum is a third proportional to the axis major and minor. PROP. XIV. The area of all parallelograms, formed by drawing tangents at the extremities of two conjugate diameters, is constant, each being equal to the rectangle under the axes. Let Pp, Dd, be any two conjugate diameters, WwXx, a parallelogram inscribed between the opposite and conjugate hyperbolas by drawing tangents at P, p, D, d; then Pp, Dd, divide the parallelogram WrXw into four equal parallelograms. Draw Pm, dm, ordinates to the axis; PF perpendicular to Dd. Let CA meet PX in T and Wx in t; B D พ F P m Mt a CT/A Ct: CA: CA: Cm, Prop. 11. CT Ct :: Cm : CM MT: Cm, by similar triangles. MT: Cm :: Cm : CM CM. MT = Cm (1) PROP. XV. The difference of the squares of any two conjugate diameters, is equal to the same constant quantity, namely, the difference of the squares of the two axes. That is, if Pp, Dd, be any two conjugate diame The rectangle under the focal distances of any point, is equal to the square of the semi-conjugate. That is, if CD be conjugate to CP, SP. HP CD. But AC. by Prop. 6. ... SP. HP: SY.HZ:: AC2: PF :: CD2: CB, by Prop. 14. CD' PROP. XVII. If two tangents be drawn, one at the principal vertex, the other at the vertex of any other diameter, each meeting the other's diameter produced, the two tangential triangles thus formed will be equal. |