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Home » Find the point in xy-plane which is equidistant from the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).
Find the point in xy-plane which is equidistant from the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).
CBSE | Class 11 | Exercise 26B | Maths | RS Aggarwal | Three Dimensional Geometry
Find the point in xy-plane which is equidistant from the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).

Answer:

The general point on xy plane is D(x, y, 0).

Consider this point is equidistant to the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).

∴ AD = BD

\sqrt {{{(x - 2)}^2} + {{(y - 0)}^2} + {{(0 - 3)}^2}} = \sqrt {{{( x - 0 )}^2} + {{(y - 3)}^2} + {{(0 - 2)}^2}}

Squaring on both sides,

(x – 2)2+ (y – 0)2 + (0 – 3)2 = (x – 0)2 + (y – 3)2 + (0 – 2)2

x2 – 4x +4 + y2 + 9 = x2 + y2 -6y + 9 + 4

-4x = -6y ….(1)

AD = CD

\sqrt {{{(x - 2)}^2} + {{(y - 0)}^2} + {{(0 - 3)}^2}} = \sqrt {{{( x - 0 )}^2} + {{(y - 0)}^2} + {{(0 - 1)}^2}}

Squaring on both sides,

(x – 2)2+ (y – 0)2 + (0 – 3)2 = (x – 0)2 + (y – 0)2 + (0 – 1)2

x2 – 4x +4 + y2 + 9 = x2 + y2 +1

-4x = -12 ….(2)

Solving equation (1) and (2),

x = 3, y = 2.

The point which is equidistant to the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1) is (3, 2, 0).

 

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