Find the coordinates of a point equidistant from four points O \left(\right. 0 , 0 , 0 \left.\right) , A \left(\right. ℓ , 0 , 0 \left.\right) , B \left(\right. 0 , m , 0 \left.\right) and C \left(\right. 0 , 0 , n \left.\right).

Engineering

Mathematics

Line of Greatest Slope

Question

Find the coordinates of a point equidistant from four points

O (0, 0, 0), A (ℓ, 0, 0), B (0, m, 0)

and

C (0, 0, n)

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

P (x, y, z)

\begin{array}{l} P O = \sqrt{x^{2} + y^{2} + z^{2}}, \\ P A = \sqrt{{(l - x),}^{2} + {(- y),}^{2} + {(- z),}^{2}} = \sqrt{{(l - x),}^{2} + y^{2} + z^{2}}, \\ P B = \sqrt{x^{2} + {(m - y),}^{2} + z^{2}}, \\ P C = \sqrt{x^{2} + y^{2} + {(n - z),}^{2}}, \\ P O = P A \Rightarrow \sqrt{x^{2} + y^{2} + z^{2}} = \sqrt{{(l - x),}^{2} + y^{2} + z^{2}}, \\ \Rightarrow x^{2} + y^{2} + z^{2} = {(l - x)}^{2} + y^{2} + z^{2} \\ \Rightarrow x = \pm (l - x), \\ \Rightarrow x = l - x \\ \Rightarrow 2 x = l \\ \Rightarrow x = \frac{l}{2}, \\ s i m i l i a r l y, y = \frac{m}{2}, z = \frac{n}{2}, \\ P (\frac{l}{2}, \frac{m}{2}, \frac{n}{2},), \end{array}

Please subscribe our Youtube channel to unlock this solution.