Fundamental Problems of Analytical Geometry

a) Given an equation, to find the corresponding locus.

b) Given a locus under some geometrical condition to determine the corresponding equation.

The following properties of the curve will be very helpful in determining the full form of locus equation.

i) Intercept: The intercept of a curve are the directed distances from the origin to the point where the curve cuts coordinates axes.

Fig (1) Fig (2)

Fig (3) Fig (4)

In the figure, OA and OB are the intercepts.

a = OA = intercept on x-axis, b = OB = intercept on y-axis.

In figure (1), 'a' and 'b' are both positive.

In figure (2), 'a' is negative and 'b' is positive.

In figure (3), 'a' and 'b' are both negative.

In figure (4), 'a' is positive and 'b' is negative.

Example:

A point (x, y) moves in such a way that its distance from A (3, -2) is always 6. Find the locus.

Suggested answer: