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

Solution:

Let A(4, 0, 0) and 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 distance formula,

Distance = √(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²

Using this,

PQ = √(4 - x)² + (0 - y)² + (0 - z)²

Calculating PB:

P (x, y, z) and B (- 4, 0, 0)

By using the distance formula,

Distance = √(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²

Using this,

PQ = √(- 4 - x)² + (0 - y)² + (0 - z)²

Since, PA + PB = 10

PA = 10 - PB

On squaring both the sides, we get

PA² = (10 - PB)²

PA² = 100 + PB² - 20PB

Therefore,

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

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

20PB = 16x + 100

Dividing both sides by 4,

5PB = (4x + 25)

On squaring both the sides again, we get

25PB² = 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

Thus, the required equation is 9x² + 25y² + 25z² - 225 = 0

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NCERT Solutions Class 11 Maths Chapter 12 Exercise 12.2 Question 5

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

Summary:

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 is 9x² + 25y² + 25z² - 225 = 0