If the point (x,y) is equidistant from the points (2p +q,2q - p) and (2p - q,p + 2q), prove - Maths - Coordinate Geometry

Question

If the point (x,y) is equidistant from the points (2p +q,2q - p)
and (2p - q,p + 2q), prove that qx = py

Mohil · Accepted Answer

he given points are (2p + q, 2q – p) and (2p – q, p + 2q) which are equidistant from (x, y).

$n n ∴ n n {n x n - n n (n 2 n p n + n q n n n^{n} n n n 2 n n} n + n n {n y n - n n (n 2 n q n - n p n) n n n^{n} n n n 2 n n} n = n {n x n - n n (n 2 n p n - n q n n n^{n} n n n 2 n n} n + n {n y n - n n (n p n + n 2 n q n n n^{n} n n n 2 n n} n n n n$

⇒ 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

If the point (x,y) is equidistant from the points (2p +q,2q - p) and (2p - q,p + 2q), prove that qx = py.