Coordinate Systems and Parametric Equations Part 2 TANGENTS & NORMALS

1. Coordinate Systems and Parametric Equations Part 2 TANGENTS & NORMALS

2. TANGENTS AND NORMALS EXAMPLE 5 A parabola has equation y2 = 9x. Find an equation for the tangent to the parabola at the point P(4, -6). y At P, O x

3. In general… A parabola has equation y O Where the sign of the gradient is the same as the sign of y. x

4. EXAMPLE 6 A rectangular hyperbola has equation xy = 18. Find the gradient of the normal to this curve when x = 6.

5. In general… A hyperbola has equation y O x

6. EXAMPLE 7 Express, in terms of t, the gradient of The parabola with equation y2 = 4ax, where a > 0 at the point P(at2, 2at), t > 0. The rectangular hyperbola with equation xy = c2 at the point , t ≠ 0. (a) A parabola has gradient

7. EXAMPLE 7 Express, in terms of t, the gradient of The parabola with equation y2 = 4ax, where a > 0 at the point P(at2, 2at), t > 0. The rectangular hyperbola with equation xy = c2 at the point , t ≠ 0. (b) A hyperbola has gradient

8. EXAMPLE 8 The diagram shows a rectangular Hyperbola, C, with equation xy = c2, For c a constant. Point P(xo, yo) is a point on C such that The line OP makes an angle α with The horizontal. The tangent, T, to the curve at P crosses The x- and y-axes at point P and Q respectively. Show that an equation for T is given by Hence, or otherwise, find the coordinates of Q and R. Show that Prove that triangles OPQ and OPR have equal areas. y α O x

9. (a) A rectangular hyperbola has gradient c2 = xoyo At P(xo, yo) x = xo Using y = mx + c At P(xo, yo)

10. (b) Hence, or otherwise, find the coordinates of Q and R. At R, x = 0, y = R y At Q, y = 0, x = Q α (c) Show that O x

11. (d) Prove that triangles OPQ and OPR have equal areas Area of Triangle OPQ = y Area of Triangle OPR = Area of Triangle OPQ = Area of Triangle OPR O x