A birdlover at $p (x, y, z)$ from a house watches two birds sitting on the branches of another tree at $A (2, 5, 8)$ and $B (3, 7, 2)$ such that AP=BP , show that $2 x + 4 y - 2 z + 31 = 0$

Open in App

Solution

Given , $A P = B P$

$\Rightarrow A P^{2} = B P^{2}$

$\Rightarrow (x - 2)^{2} + (y - 5)^{2} + (z - 8)^{2} = (x - 3)^{2} + (y - 7)^{2} + (z - 2)^{2}$

$[∵ d i s t a n c e b e t w e e n t w o p o i n t s (x_{1}, y_{1}, z_{1}) a n d (x_{2}, y_{2}, z_{2}) i s \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}]$

$\Rightarrow x^{2} + 4 - 4 x + y^{2} + 25 - 10 y + z^{2} + 64 - 16 z$

$= x^{2} + 9 - 6 x + y^{2} + 49 - 14 y + x^{2} + 4 - 4 z$

$∴ 2 x + 4 y - 12 z + 31 = 0$

Suggest Corrections

Similar questions

2 x - 2 y + z + 12 = 0

x^{2} + y^{2} + z^{2} - 2 x - 4 y + 2 z - 3 = 0

x + 4 y - 2 z = 3

3 x + y + 5 z = 7

2 x + 3 y + z = 5

\frac{2 y + 2 z - x}{a} = \frac{2 z + 2 x - y}{b} = \frac{2 x + 2 y - z}{c},

\frac{x}{2 b + 2 c - a} = \frac{y}{2 c + 2 a - b} = \frac{z}{2 a + 2 b - c} .

Join BYJU'S Learning Program