Question:
The ratio in which the $xy$-plane divides the line segment joining the points $(2, 4, 5)$ and $(3, 5, -4)$ is:
The ratio in which the $xy$-plane divides the line segment joining the points $(2, 4, 5)$ and $(3, 5, -4)$ is:
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If the coordinates of the two points have opposite signs for the coordinate perpendicular to the plane (here, $z$), the plane lies between them, meaning the division is always internal.
Updated On: May 31, 2026
- $5 : 4$ internally
- $5 : 4$ externally
- $4 : 5$ internally
- $4 : 5$ externally
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The plane $z = 0$ (the $xy$-plane) divides the line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in the ratio $-z_1 : z_2$.
Step 2: Meaning
For the points $P(2, 4, 5)$ and $Q(3, 5, -4)$, we have $z_1 = 5$ and $z_2 = -4$.
Step 3: Analysis
Using the ratio formula: \[ \text{Ratio} = -\frac{z_1}{z_2} = -\frac{5}{-4} = \frac{5}{4} = 5 : 4 \] Since the ratio is positive, the division is internal.
Step 4: Conclusion
Therefore, the $xy$-plane divides the segment in the ratio $5 : 4$ internally. Final Answer: (A)
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