1. Conic Sections Imagine you slice through a cone at different angles circle ellipse parabola rectangular hyperbola You could get a cross-section which is a: These shapes are all important functions in Mathematics and occur in fields as diverse as the motion of planets to the optimum design of a satellite dish. In FP1 you consider the algebra & geometry of 2 of these – the parabola and rectangular hyperbola

2. The Parabola Q is the point horizontally in line horizontally with P on the line x = - a The restriction that P can move such that QP = PS is the focus-directrix property The locus of points for P is a parabola P Q S(a,0) x = - a P can move such that QP=PS … Consider a point P that can move according to a rule: The point S has coordinates (a,0) The point S(a,0) is called the focus The line x = - a is called the directrix The Cartesian equation is y 2 = 4 ax

3. Sub in Figure 1 WB14 Figure 1 shows a sketch of the parabola C with equation (a) The point S is the focus of C. Find the coordinates of S. (b) Write down the equation of the directrix of C. Figure 1 shows the point P which lies on C, where y > 0, and the point Q which lies on the directrix of C. The line segment QP is parallel to the x-axis. (c) Given that the distance PS is 25, write down the distance QP, (d) find the coordinates of P, (e) find the area of the trapezium OSPQ. where the focus is S( a,0) and the directrix has equation x = - a Coordinates of S are (9,0) Equation of directrix x = -9 Focus-directrix property: PS = PQ QP = 25 Coordinates of P are (16,24)

4. WB15 Figure 1 shows a sketch of part of the parabola with equation y 2 = 12x. The point P on the parabola has x-coordinate The points A and B lie on the directrix of the parabola. The point A is on the x-axis and the y-coordinate of B is positive. Given that ABPS is a trapezium, (b) calculate the perimeter of ABPS. Figure 1 The point S is the focus of the parabola. (a) Write down the coordinates of S. where the focus is S( a,0) Directrix has equation x = - a Coordinates of S are (3,0) Sub in Focus-directrix property Perimeter = at P

5. Eg a curve has parametric equations, Parametric functions Some simple-looking curves are hard to describe with a Cartesian equation. Parametric equations, where the values of x and y are determined by a 3 rd variable t, can be used to produce some intricate curves with simple equations. t -3-20123 x y -6 Complete the table and sketch the curve 9 -4 4 -2 1 0 0 2 1 4 4 6 9 NB: you can still find the Cartesian equation of a function defined parametrically… Sub in

6. Problem solving with parametric functions Eg a curve has parametric equations, The curve meets the x-axis at A and B, find their coordinates AB At A and B, Coordinates are (-3,0) and (1,0) Eg a curve has parametric equations, The line meets the curve at A. Find the coordinates of A Substitute the expressions for x and y in terms of t to solve the equations simultaneously Solve Substitute value of t back into expressions for x and y Find values of t at A and B Substitute values of t back into expression for x

7. The parametric form of a parabola is, Does this fit with its Cartesian equation? Sub into which is true! Exam questions sometimes involve the parabola’s parametric form… WB16 The parabola C has equation y 2 = 20x. (a) Verify that the point P(5t 2,10t) is a general point on C. The point A on C has parameter t = 4. The line l passes through A and also passes through the focus of C. (b) Find the gradient of l. Sub in has focus S( a,0)

8. The equation of the straight line with gradient m that passes through is (b) Show that the equation of the tangent to C at P(12t 2, 24t) is x − ty + 12t 2 = 0. The tangent to C at the point (3, 12) meets the directrix of C at the point X. (c) Find the coordinates of X WB17 The parabola C has equation y 2 = 48x. The point P(12t 2, 24t) is a general point on C. (a) Find the equation of the directrix of C. where the focus is S( a,0) and the directrix has equation x = - a Equation of directrix x = -12 Sub at P Giving tangent Comparing (3,12) with (12t 2, 24t)at (3,12) Sub n equation of tangent When this intersects directrix x = -12 Coordinates of X are (-12,-18) Directrix

9. The Rectangular Hyperbola The rectangular hyperbola also has a focus-directrix property, but it is beyond the scope of FP1. You only need to know that: The Cartesian equation is xy = c 2 The parametric form of a parabola is, Problems involving rectangular hyperbola usually require to find the equation of the tangent or normal for functions given explicitly or in terms of c Sub

10. WB19 The rectangular hyperbola H has equation xy = c 2, where c is a positive constant. The point A on H has x-coordinate 3c. (a) Write down the y-coordinate of A. (b) Show that an equation of the normal to H at A is with general point Sub at A The equation of the straight line with gradient m that passes through is Giving normal (c) The normal to H at A meets H again at the point B. Find, in terms of c, the coordinates of B. Sub in Solve and simultaneously to find points of intersection at B Given solution x = 3 c Coordinates of B are using

11. WB20 The point P, t ≠ 0, lies on the (a) Show that an equation for the tangent to H at P is (b) The tangent to H at the point A and the tangent to H at the point B meet at the point (−9, 12). Find the coordinates of A and B. rectangular hyperbola H with equation xy = 36. The equation of the straight line with gradient m that passes through is Sub at P Giving tangent Sub in

12. WB18 The rectangular hyperbola H has equation xy = c 2, where c is a constant. (a) Show that the tangent to H at P has equation t 2 y + x = 2ct. The point Pis a general point on H. The tangents to H at the points A and B meet at the point (15c, –c). (b) Find, in terms of c, the coordinates of A and B. The equation of the straight line with gradient m that passes through is Sub at P Giving tangent Sub in Sub values in

13. Parabola Rectangular hyperbola Standard form Parametric form Foci Directrices Not required Formulae sheet facts Sub Obtaining the gradient as a function of t Parabola Rectangular hyperbola