The point in the x y - plane which is equidistant from \left(\right. 2 , 0 , 3 \left.\right) , \left(\right. 0 , 3 , 2 \left.\right) and \left(\right. 0 , 0 , 1 \left.\right) is

Engineering

Mathematics

Line of Greatest Slope

Question

The point in the

x y -

plane which is equidistant from

(2, 0, 3), (0, 3, 2)

and

(0, 0, 1)

(- 3, 2, 0)

(3, - 2.0)

(3, 2, 0)

(3, 2, 1)

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Let the points are

A (2, 0, 3), B (0, 3, 2)

and

D (0, 0, 1)

We know that Z-coordinate of every point an xy-plane is zero so let

p (x, y, 0)

be a point on xy-plane such that

P A = P B = P C

Now,

P A = P B

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

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

\Rightarrow 4 x - 6 y = 0 \Rightarrow 2 x - 3 y = 0

......(i)

and ,

P B = P C

\Rightarrow P B^{2} = P C^{2}

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

\Rightarrow - 6 y + 12 = 0

\Rightarrow y = 2

......(ii)

Putting

y = 2

in equation (i), we get

x = 3

Hence, the required point is

(3, 2, 0)

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