Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped right circular cone. Depending on the angle the plane makes with the axis of the cone, the section is a circle, an ellipse, a parabola or a hyperbola; planes through the vertex give degenerate conics. These curves appear in planetary orbits, which are ellipses, in satellite dishes and car headlights, which are parabolic, and in cooling towers. The chapter derives their standard equations. A circle with centre (h, k) and radius r has equation (x - h)^2 + (y - k)^2 = r^2. You will study the parabola y^2 = 4ax with its focus, directrix, axis and latus rectum, and its other standard forms. You will also study the ellipse x^2/a^2 + y^2/b^2 = 1 with c^2 = a^2 - b^2, and the hyperbola x^2/a^2 - y^2/b^2 = 1 with c^2 = a^2 + b^2. For each, you will learn to find the foci, vertices, lengths of axes, eccentricity e = c/a and the length of the latus rectum.

Detailed Worksheet: Conic Sections Section A - Definitions (10 marks) 1. Define a parabola in terms of focus and directrix. (2 marks) 2. Define an ellipse and its eccentricity. (2 marks) 3. Find the equation of the circle with centre (-2, 3) and radius 4. (2 marks) 4. Find the coordinates of the focus, the equation of the directrix and the length of the latus rectum of the parabola y^2 = 12x. (2 marks) 5. Find the eccentricity of the hyperbola x^2/16 - y^2/9 = 1. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the centre and radius of the circle x^2 + y^2 + 8x + 10y - 8 = 0. (3 marks) 7. Find the equation of the circle passing through (4, 1) and (6, 5) whose centre lies on the line 4x + y = 16. (3 marks) 8. For the ellipse x^2/36 + y^2/16 = 1, find the foci, vertices, lengths of the major and minor axes, eccentricity and length of the latus rectum. (3 marks) 9. Find the equation of the ellipse with vertices (13, 0) and (-13, 0) and foci (5, 0) and (-5, 0). (3 marks) 10. Find the equation of the hyperbola with vertices (0, 3) and (0, -3) and foci (0, 5) and (0, -5). (3 marks) Section C - Diagrams (10 marks) 11. Draw a double-napped cone and show the planes that produce a circle, an ellipse, a parabola and a hyperbola. (4 marks) 12. Sketch the parabola x^2 = -16y and mark its focus, directrix and latus rectum. (3 marks) 13. Sketch the ellipse x^2/25 + y^2/9 = 1 and mark its foci, vertices and the ends of the minor axis. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A parabolic reflector is 20 cm in diameter and 5 cm deep. Find the position of its focus. (5 marks) 15. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end. (5 marks) 16. Find the equation of the hyperbola with foci (0, 13) and (0, -13) and the length of the conjugate axis 24. Find its eccentricity and the length of its latus rectum. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw neat sketches with axes labelled. Show all steps.