Question:
If \(P = (3, 12, 4)\) and \(Q\) is a point on the line \(OP\) such that \(OQ = 3\), then the sum of all the coordinates of \(Q\) is
If \(P = (3, 12, 4)\) and \(Q\) is a point on the line \(OP\) such that \(OQ = 3\), then the sum of all the coordinates of \(Q\) is
Show Hint
For a point on a line through origin, use parametric form \(Q = tP\). Solve for t using distance, then compute sum of coordinates.
Updated On: Jul 18, 2026
- \(\pm \frac{10}{13}\)
- \(\pm \frac{28}{13}\)
- \(\pm \frac{19}{13}\)
- \(\pm \frac{57}{13}\)
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The Correct Option is D
Solution and Explanation
Step 1: Line equation OP.
Point P = (3,12,4), O = (0,0,0). Parametric form: \(Q = tP = (3t,12t,4t)\)
Step 2: Distance OQ = 3.
\[ |OQ| = \sqrt{(3t)^2 + (12t)^2 + (4t)^2} = \sqrt{9t^2 + 144t^2 + 16t^2} = \sqrt{169t^2} = 13|t| \]
Step 3: Solve for t.
\[ 13|t| = 3 \implies |t| = \frac{3}{13} \implies t = \pm \frac{3}{13} \]
Step 4: Coordinates of Q.
\[ Q = tP = \left(3 \cdot \frac{3}{13}, 12 \cdot \frac{3}{13}, 4 \cdot \frac{3}{13}\right) = \left(\frac{9}{13}, \frac{36}{13}, \frac{12}{13}\right) \]
Step 5: Sum of coordinates.
\[ \frac{9}{13} + \frac{36}{13} + \frac{12}{13} = \frac{57}{13} \]
Step 6: Include negative value.
Since t = -3/13 also possible, sum = \(\pm 57/13\)
Step 7: Final conclusion.
Hence, sum of coordinates of Q is \[ \boxed{\pm \frac{57}{13}} \]
Point P = (3,12,4), O = (0,0,0). Parametric form: \(Q = tP = (3t,12t,4t)\)
Step 2: Distance OQ = 3.
\[ |OQ| = \sqrt{(3t)^2 + (12t)^2 + (4t)^2} = \sqrt{9t^2 + 144t^2 + 16t^2} = \sqrt{169t^2} = 13|t| \]
Step 3: Solve for t.
\[ 13|t| = 3 \implies |t| = \frac{3}{13} \implies t = \pm \frac{3}{13} \]
Step 4: Coordinates of Q.
\[ Q = tP = \left(3 \cdot \frac{3}{13}, 12 \cdot \frac{3}{13}, 4 \cdot \frac{3}{13}\right) = \left(\frac{9}{13}, \frac{36}{13}, \frac{12}{13}\right) \]
Step 5: Sum of coordinates.
\[ \frac{9}{13} + \frac{36}{13} + \frac{12}{13} = \frac{57}{13} \]
Step 6: Include negative value.
Since t = -3/13 also possible, sum = \(\pm 57/13\)
Step 7: Final conclusion.
Hence, sum of coordinates of Q is \[ \boxed{\pm \frac{57}{13}} \]
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