Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.

Open in App

Solution

Let P (x, y, z) be any point which is equidistant from A (3,4,5) and B ( $-$ 2,1,4) .Then,
PA = PB
$\Rightarrow \sqrt{{(x - 3)}^{2} + {(y - 4)}^{2} + {(z + 5)}^{2}} = \sqrt{{(x + 2)}^{2} + {(y - 1)}^{2} + {(z - 4)}^{2}} \Rightarrow \sqrt{x^{2} - 6 x + 9 + y^{2} - 8 y + 16 + z^{2} + 10 z + 25} = \sqrt{x^{2} + 4 x + 4 + y^{2} - 2 y + 1 + z^{2} - 8 z + 16} \Rightarrow x^{2} - 6 x + 9 + y^{2} - 8 y + 16 + z^{2} + 10 z + 25 = x^{2} + 4 x + 4 + y^{2} - 2 y + 1 + z^{2} - 8 z + 16 \Rightarrow - 10 x - 6 y + 18 z + 29 = 0 ∴ 10 x + 6 y - 18 z - 29 = 0$

Hence, 10x + 6y $-$ 18z $-$ 29 = 0 is the required equation.

Suggest Corrections

Similar questions

Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.

Find the equation of the curve formed by the set of all points whose distances from the points (3, 4, -5) and (-2, 1, 4) are equal.

A (3, 4)

P

A (4, 0, 0)

B (- 4, 0, 0)

10

Join BYJU'S Learning Program