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Locus

The Circle

The Ellipse

The Parabola

The Hyperbola

General Form

Intersections of Lines & Conics

Summary&Test

 

jdsmathnotes

 

 


UNIT 9  : THE CONICS

LESSON 8: SUMMARY & REVIEW

 

LOCUS:

Definition:   A locus of points is a set of points which satisfy one or more given conditions.

 

A simple example is a circle.  A circle is a locus (set) of points each of which is a constant distance

from a fixed point.  The fixed point is the centre of the circle and the fixed distance is the radius.

 

Example 1:

a)      Determine the locus of points which are 3 units from the origin.

b)      Find the equation of the locus.

 

Solution:

a)      The locus will be a circle, centre the origin and radius of 3 units.

b)      It’s equation can be determined by following these 3 steps

 

1.      Let P(x, y) be any point on the locus (circle).

2.      Now state the condition for P to be on the locus.  P must be always 3 units

from the origin.

                       

3.      Use the distance formula to change this statement to equation form.                               

 

THE CIRCLE:

 

Definition:   A circle is a locus (set) of points each of which is a constant distance

from a fixed point.  The fixed point is the centre of the circle and the fixed distance is the radius.

 

Text Box: Main Ideas:
·	The equation of a circle, centre the origin and radius r is:
		   
·	The equation of a circle, centre (h, k) and radius r [standard form] is:
		   
·	The equation of a circle in general form is;
 
Use completing the square to put in standard form
 
 

 

 

 

 

 

 

 

 

 

 

 

 Example 2:

 Solution:

       Rewrite as follows and complete the square on both the “x” terms and the “y” terms.

 

             

 

 

THE PARABOLA:

 

Definition:   Given a fixed point F and a fixed line d in the plane.  A parabola is the locus (set) of points P in the plane, each of which is equidistant from the

 fixed point F (the focus) and the fixed line d (the directrix).  In the diagram  |PF| = |PD| for any point P on the parabola.

The vertex V is the midpoint of the perpendicular line segment from the  focus F to the directrix d .

 

 

 

 

                                                           

 

Text Box: Main Ideas for the Parabola:
·	 |PF| = |PD|  for any point P on the parabola.
·	The vertex is halfway between the focus and directrix
·	 	

·	 


·	 

·	 


Definitions:
1.	A chord is a line segment whose end points are on the curve.             		
2.	A focal chord is a chord that passes through the focus.
3.	The focal length is the distance from the vertex to the focus [VF]
4.   The length of the focal chord perpendicular to the 
axis of symmetry of the parabola is called the
focal width.
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Example 3:

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

THE ELLIPSE:

 

Definition:   Given two fixed points in the plane F1 and F2.  An ellipse is the locus (set) of points P such that the sum |PF1+ PF2 |  is a constant.  The two fixed points are the foci.

The centre is the midpoint of the line segment joining the two foci (F1F2).

 

Text Box: Main Ideas:
·	The sum |PF1+ PF2 | = constant = 2a 
·	A2A1 is the Major Axis and  |A2A1| = 2a
·	B2B1 is the Minor Axis and  |B2B1| = 2b
	
·	The standard form of the equation of an ellipse, centre (0, 0), foci on the x-axis:is:
		                             
·	The standard form of the equation of an ellipse, centre (0, 0), foci on the y-axis:is:
·		                             
·	The standard form of the equation of an ellipse, centre (h, k), major axis parallel to the x-axis:is:
·	
·	                           
·	The standard form of the equation of an ellipse, centre (h,k), major axis parallel to the y-axis:is:
·	
·	                           
·	                              
·	Note a > b always for the ellipse.
·	For the ellipse the Pythagorean relation  a2 = b2 + c2  holds true.
 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 4:

 

 

 

 

Solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 THE HYPERBOLA :

 

 

Definition:   Given two fixed points in the plane F1 and F2.  A hyperbola is the locus (set) of points P such that the difference |PF2 – PF1 |  is a constant.  The two fixed points are the foci.

The centre is the midpoint of the line segment joining the two foci (F1F2).

 

 

 

 

 

 

 

Text Box: Main Ideas: See diagrams at left and above.
·	The sum |PF2 -  PF1 | = constant = 2a 
·	A2A1 is the Transverse Axis and  |A2A1| = 2a
·	B2B1 is the Conjugate Axis and  |B2B1| = 2b
	
·	The standard form of the equation of a hyperbola, centre (0, 0), foci on the x-axis:is:
		                              
·	The standard form of the equation of a hyperbola, centre (0, 0), foci on the y-axis:is:
·	
	                              
·	The standard form of the equation of a hyperbola, centre (h, k), major axis parallel to the x-axis:is:
·	
                            
·	The standard form of the equation of an ellipse, centre (h,k), major axis parallel to the y-axis:is:
                                 
·	                               
·	For the hyperbola the Pythagorean relation  c2 = a2 + b2  holds true.
 
       

·
 

 

 


 

 Hyperbola with foci on the x-axis:

 

 

Hyperbola with foci on the y-axis:

 

 

 

 

 

 

 

 Example 5:

 

Given the hyperbola with equation 9x2 –54x – 25y2 + 200y – 544 = 0.

a) Put the equation in standard form and find the

    values of a, b, c, e.

b) Determine the coordinates of the vertices.

c) State the length of the transverse and conjugate axes.

d) Find the equations of the asymptotes.

e) Draw the graph

 

 

 

 

                 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

GENERAL FORM OF THE EQUATION OF A CONIC: ax2 + by2 +2gx + 2fy + c = 0

                                                                            

The type of conic can be determined by the values of the coefficients “a and b” in the equation according to the following chart.

 

                                               

Type of Conic

            Condition on a, b

Circle

a = b

Parabola

ab = 0

Ellipse

ab > 0

Hyperbola

ab < 0

 

 

 

INTERSECTION OF LINES AND CONICS:

 

Example 6:

Find the point(s) of intersection of the line  x = y – 5 = 0 and the conic with equation y2 – 11x2 = 5.

 

 

 

 

 

 

  

 

    

 

 

 

 

 

   

 

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