(b) Explain why AB BA. ≠ (4) (ii) Given that , where is a real number k find C–1, giving your answer in terms of k. (3) 5. (a) Use the standard results for and to show that (6) (b) Hence show that where and are constants to be found. a b (3) The rectangular hyperbola has cartesian equation = 6. H xy c2. The point > 0, is a general point on P, t H. (a) Show that an equation of the tangent to at the point is H P + = 2ct t2y x (4) An equation of the normal to at the point is = H P t3x ty ct4 c. – – Given that the normal to at meets the at the point and the tangent to at H P x-axis A H P meets the at the point x-axis B, (b) find, in terms of and the coordinates of and the coordinates of c t, A B. (2) Given that = 4, c (c) find, in terms of the area of the triangle Give your answer in its simplest form. t, APB. (3)
(b) Explain why AB ≠ BA. (4) (ii) Given that , where k is a real number find C–1, giving your answer in terms of k. (3) 5. (a) Use the standard results for and to show that (6) (b) Hence show that where a and b are constants to be found. (3) 6. The rectangular hyperbola H has cartesian equation xy = c2. The point P, t > 0, is a general point on H. (a) Show that an equation of the tangent to H at the point P is t2y + x = 2ct (4) An equation of the normal to H at the point P is t3x – ty = ct4 – c. Given that the normal to H at P meets the x-axis at the point A and the tangent to H at P meets the x-axis at the point B, (b) find, in terms of c and t, the coordinates of A and the coordinates of B. (2) Given that c = 4, (c) find, in terms of t, the area of the triangle APB. Give your answer in its simplest form. (3)
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