Q. 54.1( 101 Votes )

Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (– 4, 0, 0) is equal to 10.

Answer :

Let A (4, 0, 0) & B (– 4, 0, 0)


Let the coordinates of point P be (x, y, z)


Calculating PA


P ≡ (x, y, z) and A ≡ (4, 0, 0)


Distance PA


Here,


x1 = x, y1 = y, z1 = z


x2 = 4, y2 = 0, z2 = 0


Distance PA


Calculating PB


P ≡ (x, y, z) and B ≡ (– 4, 0, 0)


Distance PB


Here,


x1 = x, y1 = y, z1 = z


x2 = – 4, y2 = 0, z2 = 0


Distance PB


Given that –


PA + PB = 10


PA = 10 – PB


Squaring both sides, we get –


PA2 = (10 – PB)2


PA2 = 100 + PB2 – 20 PB


(4 – x)2 + (0 – y)2 + (0 – z)2


= 100 + (– 4 – x)2 + (0 – y)2 + (0 – z)2 – 20 PB


(16 + x2 – 8x) + (y2) + (z2)


= 100 + (16 + x2 + 8x) + (y2) + (z2) – 20 PB


20 PB = 16x + 100


5 PB = (4x + 25)


Squaring both sides again, we get –


25 PB2 = 16x2 + 200x + 625


25 [(– 4 – x)2 + (0 – y)2 + (0 – z)2] = 16x2 + 200x + 625


25 [x2 + y2 + z2 + 8x + 16] = 16x2 + 200x + 625


25x2 + 25y2 + 25z2 + 200x + 400 = 16x2 + 200x + 625


⇒ 9x2 + 25y2 + 25z2 – 225 = 0


This is the required equation.

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