If the point (x,y) is equidistant from the points (2p +q,2q - p) and (2p - q,p + 2q), prove that qx = py.
he given points are (2p + q, 2q – p) and (2p – q, p + 2q) which are equidistant from (x, y).
⇒ x2 + (2p + q)2 –2x (2p + q) + y2 + (2q – p)2 –2y (2q – p) = x2 + (2p – q)2 –2x (2p – q) + y2 + (p + 2q)2 –2y(p + 2q)
⇒ (2p + q)2 – 2x (2p + q) + (2q – p)2 – 2y (2q – p) = (2p – q)2 –2x (2p – q) + (p + 2q)2 – 2y (p + 2q)
= 4p2 + q2 + 4pq – 4xp – 2xq + 4q2 + p2 – 4pq – 4yq + 2yp = 4p2 + q2– 4pq – 4xp + 2xq + p2 + 4q2 + 4pq – 2yp – 4qy
⇒ – 4xq = – 4yp
⇒ xq = yp
⇒ xq = py