Writing x = ar³, y = ar⁹ and z = ar¹⁵, we compute y² = (ar⁹)² = a²r¹⁸. Also x x z = (ar³) x (ar¹⁵) = a²r¹⁸. Since y² = x x z, x, y, z form a GP.
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Squaring x + y + z = 0 gives x2 + y2 + z2 + 2(xy + yz + zx) = 0. Since xy + yz + zx = 0, it simplifies to x2 + y2 + z2 = 0. ...
Yes, the points coincide only when the x-coordinate is equal to the y-coordinate. For example, if x = 5 and y = 5, both points P and Q become (5, 5).
Use the midpoint formula: x-mid = (x1 + x2) / 2. For x: –7 = (3 + x) / 2 which gives x = –17. For y: 1 = (–4 + y) / 2 which gives y = 6. Point ...