Two watch-towers are located at and in a forest area represented on a coordinate grid
MATHEMATICS · Multi-part · A Level · Mathematics · Coordinate geometry in the (x y) plane · Coordinate geometry in the (x y) plane
Problem
Two watch-towers are located at W1(−4,5) and W2(6,−3) in a forest area represented on a coordinate grid. A straight east–west highway is represented by the line y=−1. A ranger wishes to set up a small supply base on this highway such that the base is equidistant from both watch-towers.
(i) Let the coordinates of the supply base be S(x,−1). Write the equality of distances that must hold so that S is equidistant from W1 and W2 without substituting any specific values.
(ii) Solve the equation obtained in part (i) to find the coordinates of S.
(iii) Determine the common distance from each watch-tower to the supply base, giving your answer correct to one decimal place.
(iv) Explain whether more than one such point S can exist on the highway y=−1 that is equidistant from W1 and W2, and justify your answer.