Question

If the sum of the squares of the distances of a point \(P(x, y, z)\) from the three co-ordinate axes is \(324\), then the distance of point \(P\) from the origin is ....

• A. \(18\)
• B. \(162\)
• C. \(9\sqrt{2}\)
• D. \(324\)

Hint:

If a point is involved with distances from coordinate axes, directly use: \[ d_x^2=y^2+z^2,\quad d_y^2=z^2+x^2,\quad d_z^2=x^2+y^2 \]

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Distances from axes to \(P(x, y, z)\) are \(\sqrt{y^2+z^2}\), \(\sqrt{x^2+z^2}\), and \(\sqrt{x^2+y^2}\).
Step 2: Key Formula or Approach:
Sum of squares \(= (y^2+z^2) + (x^2+z^2) + (x^2+y^2) = 2(x^2+y^2+z^2)\).
Step 3: Detailed Explanation:
Given \(2(x^2+y^2+z^2) = 324 \implies x^2+y^2+z^2 = 162\).
Distance from origin \(= \sqrt{x^2+y^2+z^2} = \sqrt{162} = \sqrt{81 \times 2} = 9\sqrt{2}\).
Step 4: Final Answer:
Distance is \(9\sqrt{2}\).