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.

Asked by Abhisek | 11 months ago |  114

Solution :-

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)

By using the formula,

Distance PA =

$$\sqrt{(x^2 – x^1)^2 + (y^2 – y^1)^2 + (z^2 – z^1)^2}$$

So here,

x1 = x, y1 = y, z1 = z

x2 = 4, y2 = 0, z2 = 0

Distance PA =

$$\sqrt{(4– x)^2 + (0 – y)^2 + (0 – z)^2}$$

Calculating PB

P ≡ (x, y, z) and B ≡ (– 4, 0, 0)

By using the formula,

Distance PB

$$\sqrt{(x^2 – x^1)^2 + (y^2 – y^1)^2 + (z^2 – z^1)^2}$$

So here,

x1 = x, y1 = y, z1 = z

x2 = – 4, y2 = 0, z2 = 0

Distance PB

$$\sqrt{(4– x)^2 + (0 – y)^2 + (0 – z)^2}$$

Now it is given that:

PA + PB = 10

PA = 10 – PB

Square on both the 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)

Square on both the 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

The required equation is 9x2 + 25y2 + 25z2 – 225 = 0

Answered by Pragya Singh | 11 months ago

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