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 \(2d_2^2+d_3^2+1=\)
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 \(2d_2^2+d_3^2+1=\)
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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.
Updated On: Jul 18, 2026
- \(d_1^2\)
- \(2d_1^2\)
- \(3d_1^2\)
- \(4d_1^2\)
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The Correct Option is B
Solution and Explanation
Step 1: Recall the formula for distance from coordinate axes.
For a point \((x,y,z)\), its distance from the \(X\)-axis is \[ d_1=\sqrt{y^2+z^2} \] Its distance from the \(Y\)-axis is \[ d_2=\sqrt{x^2+z^2} \] Its distance from the \(Z\)-axis is \[ d_3=\sqrt{x^2+y^2} \]
Step 2: Calculate \(d_1^2,d_2^2,d_3^2\).
Given point is \[ (1,2,3) \] Here, \[ x=1,\quad y=2,\quad z=3 \] Therefore, \[ d_1^2=y^2+z^2 \] \[ d_1^2=2^2+3^2 \] \[ d_1^2=4+9=13 \] Similarly, \[ d_2^2=x^2+z^2 \] \[ d_2^2=1^2+3^2 \] \[ d_2^2=1+9=10 \] Also, \[ d_3^2=x^2+y^2 \] \[ d_3^2=1^2+2^2 \] \[ d_3^2=1+4=5 \]
Step 3: Evaluate the given expression.
Now, \[ 2d_2^2+d_3^2+1 \] Substitute the values, \[ 2d_2^2+d_3^2+1=2(10)+5+1 \] \[ =20+5+1 \] \[ =26 \]
Step 4: Express the result in terms of \(d_1^2\).
Since, \[ d_1^2=13 \] we get \[ 26=2(13) \] Therefore, \[ 26=2d_1^2 \] Hence, \[ 2d_2^2+d_3^2+1=2d_1^2 \]
Step 5: Final conclusion.
Therefore, \[ \boxed{2d_1^2} \]
For a point \((x,y,z)\), its distance from the \(X\)-axis is \[ d_1=\sqrt{y^2+z^2} \] Its distance from the \(Y\)-axis is \[ d_2=\sqrt{x^2+z^2} \] Its distance from the \(Z\)-axis is \[ d_3=\sqrt{x^2+y^2} \]
Step 2: Calculate \(d_1^2,d_2^2,d_3^2\).
Given point is \[ (1,2,3) \] Here, \[ x=1,\quad y=2,\quad z=3 \] Therefore, \[ d_1^2=y^2+z^2 \] \[ d_1^2=2^2+3^2 \] \[ d_1^2=4+9=13 \] Similarly, \[ d_2^2=x^2+z^2 \] \[ d_2^2=1^2+3^2 \] \[ d_2^2=1+9=10 \] Also, \[ d_3^2=x^2+y^2 \] \[ d_3^2=1^2+2^2 \] \[ d_3^2=1+4=5 \]
Step 3: Evaluate the given expression.
Now, \[ 2d_2^2+d_3^2+1 \] Substitute the values, \[ 2d_2^2+d_3^2+1=2(10)+5+1 \] \[ =20+5+1 \] \[ =26 \]
Step 4: Express the result in terms of \(d_1^2\).
Since, \[ d_1^2=13 \] we get \[ 26=2(13) \] Therefore, \[ 26=2d_1^2 \] Hence, \[ 2d_2^2+d_3^2+1=2d_1^2 \]
Step 5: Final conclusion.
Therefore, \[ \boxed{2d_1^2} \]
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