Your Task: Each of these situations describe a point P or (x,y) moving according to some conditions. First, try to determine the geometrical nature of the locus of points formed as P moves. Second, try to use either algebra or geometry to justify your claim.
Locus 1: The point P moves so that the line segment connecting it to the point (0,0) is perpendicular to the line segment connecting it to the point (4,0).
Locus 2: The point P moves so that its distance from the y-axis equals its distance from the point (6,0).
Locus 3: The point P moves so that the sum of its distances from the points (4,0) and (-4,0) equals 10.
Locus 4: The point P moves so that its distance from the point (0,9) is three times its distance from the line y = 1.
Locus 5: The point P is the midpoint of a line segment of length 6 with its ends anchored on the x- and y-axes. What is the locus of P as the lines segment moves?
Locus 6: The point P is the midpoint of the line segment connecting the point (8,0) to a point B on the circle x2 + y2= 4. What is the locus of P as point B moves around the circle?
Source: Adapted from Denbow & Goedicke's Foundations of Mathematics, 1959
Hint: Create the starting situation and then start tracing the trail of the point P as it moves. What general relationships seem to be evident?
Solution Commentary: The first is a circle, illustrating the theorem that angles inscribed in a semicircle are right angles. Can you find the equatioin of the circle?
The other loci are left for you to puzzle over...all are do-able.