DISCOVERING CONICS WITH. Dr Toh Pee Choon NIE 2 June 2016

Transcription

1 DISCOVERING CONICS WITH Dr Toh Pee Choon NIE 2 June 2016

2 Introduction GeoGebra is a dynamic mathematics software that integrates both geometry and algebra Open source and free to download Version 5.0 includes 3D capabilities and is particularly useful for conics

3 Exercise 1: Basic Concepts Use the point tool to create four points. (Note that they are automatically labelled.) Use the line tool to create a line through points A and B. (Note that when the pointer is above the tool, instructions on the sequence of clicks will appear.) Now create a segment joining B and C, and a ray from C to D. (Click the line tool to see the drop-down menu.) Use the circle tool to create a circle with centre at A and contains B.

4 Exercise 2: Simple Loci It might be helpful to use a new window for every exercise. Use the File drop-down menu. Create two points. Rename them F 1 and F 2 by rightclicking and selecting rename. Input F_1 in the dialogue box. The locus of points equidistant from F 1 and F 2 is the perpendicular bisector and can be denoted Use the perpendicular bisector tool (fourth icon group) to draw the locus

5 Exercise 3: Parabola Given a point F called the focus and a line d, called the directrix, the parabola is the locus of points equidistant from F and d. By convention we always use the shortest (i.e. perpendicular) distance for distance to a line. Create a point F and a line d. (Right-click to rename the line and note that its equation automatically appears in the Algebra window.) Use the Parabola tool (seventh icon group) to draw the locus parabola. Create a point P on the parabola. Use the Perpendicular line tool (fourth icon group) to obtain D the foot of the perpendicular from P to line d.

6 Exercise 3: Parabola (cont d) Verify PF = PD by creating the line segments PF and PD. (Note that the lengths of these two segments will appear in the algebra window. Or use distance tool from the eighth icon group.) Draw P along the parabola to see the invariance. In a new window, create the parabola y= x^2/4 by using the input dialogue at the bottom of the screen. Use the move tool (last icon group) if necessary to centralize the parabola. Create the points F=(0,1) and line d:y=-1 in the input dialogue. F is the focus and d the directrix. Verify PF = PD as above.

7 Exercise 4: Parabola (first principles) We wish to create a parabola from its focus and directrix without the tool. Create a point F=(2,0) and line d:x=-2. Every point P on the parabola corresponds one-to-one to a point on directrix. So we will use a point D on d to create our parabola. If PD = PF, we have an isosceles triangle, which means that foot of the perpendicular from P to DF is exactly at the midpoint of DF. Construct a line perpendicular to d at point D and the perpendicular bisector of D and F. P lies on the intersection of these two lines. Right-click P and set Trace on to plot the parabola.

8 Exercise 5: Reflection Property (Parabola) Create a parabola: y^2=4*x in the Input dialogue. Create the focus F=(1,0) and any point P on the parabola. The reflection property states light coming parallel to the axis of a parabolic mirror is reflected at the parabola to pass through the focus. This is equivalent to saying that the reflection of ray PF about the normal at P is parallel to the x-axis Use the tangent tool (fourth icon group) to create the tangent to the parabola at P, and the perpendicular line tool to create the normal. Construct ray PF and use the reflection tool (ninth icon group) to create the reflection. Vary P.

9 Ex 6: Geometric Property (Ellipse) The slider tool (11 th icon group) is extremely useful. It works like a variable. Create a slider and in the dialogue box, rename it as c and change minimum to 0. F_1 = (c,0) and F_2 = (-c,0) creates two points whose coordinates are governed by the variable c in the slider. Create point A=(0,3) and use the ellipse tool (7 th icon group) to create an ellipse with foci at F 1 and F 2. Vary the slider to see the change in the ellipse Verify the geometric property, i.e. if P is on the ellipse, PF 1 + PF 2 is constant. Rename the segments to m and n respectively and Input k=m+n. Note that k is constant as P varies.

10 Ex 6: Ellipse (cont d) The reflection property of an ellipse states that light from one focus is reflected in an elliptic mirror to pass through the other focus. In other words, the angle between PF 1 and the tangent line at P is the same as that of PF 2 and the tangent line. Create the tangent line at P and one point on each side of P. (The position of these points do not matter and they are merely used for calculation of angles.) The angle tool (8 th icon group) can be used to measure the two required angles. Vary P to verify invariance. (One can also change the value of c).

11 Ex 7: Geometric Property (Hyperbola) The hyperbola can be defined by the locus of P where PF 1 - PF 2 = k is constant. Create foci F 1,F 2 who are 2 units apart. Create a slider k from -5 to 5. Create a slider r from 0 to 20, r is meant to be the value PF 2 and so PF 1 =r+k. In other words, we have two circles. The intersection of the circles gives a hyperbola as r varies.

12 Ex 8: DEF definition of conics A conic (parabola, ellipse or hyperbola) can be uniformly defined by its Directrix-Eccentricity-Focus PF =e Pd Create a slider with eccentricity e between 0 and 5 Create a slider k, from -5 to 20 and directrix d:x=k Create the focus, F=(0,0). Create a slider m between 0 to 100 where m is the value of Pd. This means that P lies on one of the lines x=k+m or x=k-m. (Create these two lines) On the other hand, P lies on a circle em distance away from F. The intersections (four possibilities) of this circle with the previous two lines gives P. Use trace (vary or animate m) to visualize the conic.

13 Ex 9: 3D plots Plot z^2=x^2+y^2 in the Input Dialogue. If necessary use the View drop-down window for 3D graphics Create a slider k from 0 to 5 and the plane z-1=kx Rotate the 3D window to view the intersection which is a parabola when k=1, ellipse when k < 1 and hyperbola when k > 1. The Intersect Two Surfaces Tool (6 th icon in 3D view) can be used to find the intersection. Right-clicking the intersection will reveal an option: Create 2D view.

14 Ex 10: Discovery Exercise 1 Let P be a point on an ellipse and F 1 and F 2 be the two foci. The point M lies on the tangent line at P such that F 1 M is perpendicular to the tangent. Let Q be the intersection of the ray F 2 P and F 1 M. What do you observe about Q as P varies? State a conjecture and prove it.

15 Ex 11: Discovery Exercise 2 Consider a hyperbola with transverse axis on the x-axis. (i.e. foci are also on x-axis) Let A and B be vertices of the hyperbola Let C and D be points on the hyperbola with positive x- coordinates. Let Q be the intersection of DB and CA and R be the intersection of DA and CB. What do you observe about Q and R as the chord CD varies?

16 Ex 12: Discovery Exercise 3 Consider a hyperbola x^2-y^2=a^2 Let A and B be vertices of the hyperbola Let CD be a chord of the hyperbola such that CD is perpendicular to the x-axis. What do you observe about angles CAD and CBD as the chord CD varies?