Two students, Arjun and Beena, stand at points A(2,3) and B(10,7) respectively on a school ground that has been laid out as a coordinate grid. Their teacher asks them to choose a point on the straight path represented by the equation x=y so that each of them has to walk the same distance to reach it.
(i) If the required meeting point is P(h,h) on the path x=y, write the condition that must be satisfied so that PA=PB. Do not evaluate any distance at this stage.
(ii) Solve the condition obtained in part (i) to determine the exact coordinates of P.
(iii) Calculate the walking distance for each student from his or her starting point to P, and hence find the total distance covered by both students together.
(iv) Is there any other point on the path x=y in the first quadrant that is equidistant from A and B? Give a reason for your answer.