     Home Locus The Circle The Ellipse The Parabola The Hyperbola General Form Intersections of Lines & Conics Summary&Test # 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. 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 .  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).  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).  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.  