Question

3. Find the locus of a point, which moves so that its distance
from (1,2,3) is four times its distance from YZ-plane.​

Answer 1

We have to find the locus of a point which moves so that its distance from (1, 2, 3) is four times it's distance from YZ plane.

Solution : let (x , y, z) is the point which moves so that its distance from (1,2,3) is four times its distance from YZ plane {i.e., (0, y, z). }

distance between (x, y, z) and (1, 2, 3) = 4 × distance between (x, y, z) and (0, y, z)

⇒(x - 1)² + (y - 2)² + (z - 3)² = 4²[(x - 0)² + (y - y)² + (z - z)²]

⇒x² + y² + z² - 2x - 4y - 6z + 14 = 16x²

⇒15x² - y² - z² + 2x + 4y + 6z - 14 = 0

Therefore the locus of point is 15x² - y² - z² + 2x + 4y + 6z - 14.

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Answer 2

Answer:-

We have to find the locus of a point which moves so that its distance from (1, 2, 3) is four times it's distance from YZ plane.

Solution :

let (x , y, z) is the point which moves so that its distance from (1,2,3) is four times its distance from YZ plane {i.e., (0, y, z). }

distance between (x, y, z) and (1, 2, 3) = 4 × distance between (x, y, z) and (0, y, z)

⇒(x - 1)² + (y - 2)² + (z - 3)² = 4²[(x - 0)² + (y - y)² + (z - z)²]

⇒x² + y² + z² - 2x - 4y - 6z + 14 = 16x²

⇒15x² - y² - z² + 2x + 4y + 6z - 14 = 0

Therefore the locus of point is 15x² - y² - z² + 2x + 4y + 6z - 14.

Hope it's help you❤️