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Conic Sections

Conic Sections. Imagine you slice through a cone at different angles. You could get a cross-section which is a:. circle. ellipse. parabola. rectangular hyperbola.

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Conic Sections

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  1. Conic Sections Imagine you slice through a cone at different angles You could get a cross-section which is a: circle ellipse parabola rectangular hyperbola 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 Consider a point P that can move according to a rule: Q is the point horizontally in line horizontally with P on the line x = -a The point S has coordinates (a,0) P can move such that QP=PS … P The locus of points for P is a parabola Q The point S(a,0) is called the focus The line x = -a is called the directrix S(a,0) The restriction that P can move such that QP = PS is the focus-directrix property The Cartesian equation is y2 = 4ax x = -a

  3. 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. Figure 1 where the focus is S(a,0) and the directrix has equation x = -a Coordinates of S are (9,0) (b) Write down the equation of the directrix of C. Equation of directrix x = -9 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, Focus-directrix property: PS = PQ (e) find the area of the trapezium OSPQ. QP = 25 (d) find the coordinates of P, Sub in Coordinates of P are (16,24)

  4. WB15 Figure 1 shows a sketch of part of the parabola with equation y2 = 12x . The point P on the parabola has x-coordinate 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) Coordinates of S are (3,0) 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. Directrix has equation x = -a at P Sub in Focus-directrix property Perimeter =

  5. 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 3rd variable t, can be used to produce some intricate curves with simple equations. Eg a curve has parametric equations , Complete the table and sketch the curve -6 -4 -2 0 2 4 6 9 4 1 0 1 4 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 At A and B, Find values of t at A and B Substitute values of t back into expression for x A 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

  7. Does this fit with its Cartesian equation? The parametric form of a parabola is , Sub into which is true! Exam questions sometimes involve the parabola’s parametric form… WB16 The parabola C has equation y2 = 20x. (a) Verify that the point P(5t2 ,10t) is a general point on C. Sub in 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. has focus S(a,0)

  8. WB17 The parabola C has equation y2 = 48x. The point P(12t2, 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 (b) Show that the equation of the tangent to C at P(12t2, 24t) is x − ty + 12t2 = 0. The equation of the straight line with gradient m that passes through is at P Sub Giving tangent The tangent to C at the point (3, 12) meets the directrix of C at the point X. (c) Find the coordinates of X Directrix Comparing (3,12) with (12t2, 24t) at (3,12) Sub n equation of tangent When this intersects directrix x = -12 Coordinates of X are (-12,-18)

  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 = c2 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 = c2, where c is a positive constant. The point A on H has x-coordinate 3c. (a) Write down the y-coordinate of A. (c) The normal to H at A meets H again at the point B. Find, in terms of c, the coordinates of B. with general point Solve and simultaneously to find points of intersection (b) Show that an equation of the normal to H at A is Sub in Sub at A The equation of the straight line with gradient m that passes through is Given solution x = 3c Giving normal at B using Coordinates of B are

  11. WB20 The point P , t ≠ 0, lies on the rectangularhyperbola H with equation xy = 36. (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. at P Sub Sub in The equation of the straight line with gradient m that passes through is Giving tangent Sub in

  12. WB18 The rectangular hyperbola H has equation xy = c2, where c is a constant. The point P is a general point on H. (a) Show that the tangent to H at P has equation t2y + x = 2ct. 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. at P Sub Sub in The equation of the straight line with gradient m that passes through is Giving tangent Sub values in

  13. Formulae sheet facts Not required Not required Obtaining the gradient as a function of t Parabola Sub Rectangular hyperbola Sub

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