(6) (b) Hence show that where and are integers to be found. a, b c (4) 6. A parabola has equation = 4ax, > 0 C y2 a The points 2ap) and 2aq) lie on where 0, 0, P(ap2, Q(aq2, C, p q p q. ≠ ≠ ≠ (a) Show that an equation of the tangent to the parabola at is P = py x ap2 – (4) (b) Write down the equation of the tangent at Q. (1) The tangent at meets the tangent at at the point P Q R. (c) Find, in terms of and the coordinates of giving your answers in their simplest p q, R, form. (4) Given that lies on the directrix of R C, (d) find the value of pq. (2) 7. = 2 + 3i, = 3 + 2i, = z1 z2 z3 a + bi, a, b (a) Find the exact value of |z1 + z2|. (2) Given that = , w (b) find in terms of and giving your answer in the form iy, . w a b, x + x, y (4) Given also that = , w (c) find the value of and the value of a b, (3)
(6) (b) Hence show that where a, b and c are integers to be found. (4) 6. A parabola C has equation y2 = 4ax, a > 0 The points P(ap2, 2ap) and Q(aq2, 2aq) lie on C, where p ≠ 0, q ≠ 0, p ≠ q. (a) Show that an equation of the tangent to the parabola at P is py – x = ap2 (4) (b) Write down the equation of the tangent at Q. (1) The tangent at P meets the tangent at Q at the point R. (c) Find, in terms of p and q, the coordinates of R, giving your answers in their simplest form. (4) Given that R lies on the directrix of C, (d) find the value of pq. (2) 7. z1 = 2 + 3i, z2 = 3 + 2i, z3 = a + bi, a, b (a) Find the exact value of |z1 + z2|. (2) Given that w = , (b) find w in terms of a and b, giving your answer in the form x + iy, x, y . (4) Given also that w = , (c) find the value of a and the value of b, (3)
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