Question

If \(d_1,d_2,d_3\) are the distances of the point \((1,2,3)\) from the \(X,Y,Z\)-coordinate axes respectively, then \(2d_2^2+d_3^2+1=\)

• A. \(d_1^2\)
• B. \(2d_1^2\)
• C. \(3d_1^2\)
• D. \(4d_1^2\)

Hint:

For a point \((x,y,z)\), the distance from an axis is found by ignoring the coordinate of that axis and using the remaining two coordinates.

Answer 1

The Correct Option is B

Step 1: Write each distance-squared in terms of x, y, z.
For a point \((x,y,z)\), distance from the x-axis is \(d_1=\sqrt{y^2+z^2}\), from the y-axis is \(d_2=\sqrt{x^2+z^2}\), and from the z-axis is \(d_3=\sqrt{x^2+y^2}\).

Step 2: Build the target expression symbolically.
\[2d_2^2+d_3^2+1 = 2(x^2+z^2)+(x^2+y^2)+1 = 3x^2+y^2+2z^2+1\]
and
\[2d_1^2 = 2(y^2+z^2) = 2y^2+2z^2\]

Step 3: Compare the two at the given point.
With \((x,y,z)=(1,2,3)\):
\[2d_2^2+d_3^2+1 = 3(1)+4+2(9)+1 = 3+4+18+1=26\]
\[2d_1^2 = 2(4+9) = 2(13)=26\]

Step 4: Conclusion.
The two sides match exactly, so
\[\boxed{2d_1^2}\]