Locus

A locus (pronounced lock-us) is a set of points that satisfy a certain criteria. This may be something like a circle which is a locus of points that are equidistant from a certain point, or it could be a region that satisfies an inequality.

In the Extension 1 course it usually means using the distance formula on a variable point.

Common types of loci

Circle

A circle is a locus of points such that its distance from a point (centre) is constant.

Eg. The point P(x,y) moves such that its distance from A(1,2) is 2 units. Find the expression of the locus of P and describe it geometrically.

Distance of PA = √((x-1)²+(y-2)²) = 2

(x-1)² + (y-2)² = 4

The locus of P is a circle with a radius of 2 units and the centre at (1,2)

Straight line

A straight line is a locus of points that are equidistant from two points.

Eg. The point P(x,y) moves such that its distance from A(-1,2) and B(3,-1) are equal. Find the expression of the locus.

PA = √((x+1)² + (y-2)²)

PB = √((x-3)² + (y+1)²)

PA = PB

∴ (x+1)² + (y-2)² = (x-3)² + (y+1)²

x² + 2x + 1 + y² – 4y + 4 = x² -6x + 9 + y² +2y +1

8x – 6y – 5 = 0

Parabola

A parabola is a locus of points that is equidistant from a fixed point called the focus and a line called the directrix.

Eg. The point P(x,y) moves such that its distance from the point S(0,1) and the line y = -1 is equal. Find the locus of point P.

PS = √(x² + (y-1)²)

Distance from P to y=-1 is y + 1

∴ x² + (y-1)²= (y+1)²

x² + y² -2y + 1 = y² +2y +1

x² = 4y

y= x²/4

Ellipse

The ellipse is a locus of points that are the sum of the distances from two points (called foci) is constant.

i.e. PA + PB = c

Hyperbola

The hyperbola is a locus of points where the difference of the distances from two points (called foci) is constant.

i.e. PA – PB = c

The ellipse and hyperbola will not be studied in detail in Extension 1 Mathematics. It will be studied in Extension 2 Mathematics in a topic called Conics.