Find the point on the z−axis which is equidistant from the points A(1,5,7) and B(5,1,−4).

Engineering

Mathematics

Vector Addition and Subtraction

Distance from a Plane

Question

Find the point on the z−axis which is equidistant from the points A(1,5,7) and B(5,1,−4).

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Given that

A = (1,5,7) B = (5,1,4)

To Find - A point at z axis, equidistant from A and B.

Consider the point be P having co- ordinates (0,0,a)

By Question

$\vec{PA} = \vec{PB}$

(PA)²= (PB)²

⇒ (1 – 0)²+ (5 – 0)²+(7 – a)²= (5 – 0)²+ (1 – 0)²+ (–4 – a)²

⇒ 1 + 25 + 49+a²– 14a = 25 + 1 + 16 + a²+ 8a

⇒ 14a + 8a = 49 – 16

⇒ 22a = 33

⇒ a = 3/2

Therefore point 'p' is (0, 0,3/2).