Question:
The equation of the plane passing through the midpoint of the line joining the points (1,2,3) and (3,4,5) and perpendicular to it is
The equation of the plane passing through the midpoint of the line joining the points (1,2,3) and (3,4,5) and perpendicular to it is
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This plane is the "Perpendicular Bisector Plane." Normal = $\vec{AB}$, Point = Midpoint.
Updated On: Jul 3, 2026
- $x + y + z = 9$
- $x + y + z + 9 = 0$
- $2x + 3y + 4z = 9$
- $2x + 3y + 4z + 9 = 0$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The normal to the plane is the vector joining the two points. The plane passes through the midpoint.
Step 2: Meaning
Midpoint $M = (\frac{1+3}{2}, \frac{2+4}{2}, \frac{3+5}{2}) = (2, 3, 4)$. Direction ratios of the normal $\vec{n} = (3-1, 4-2, 5-3) = (2, 2, 2)$, which is equivalent to $(1, 1, 1)$.
Step 3: Analysis
Equation of plane: $1(x - 2) + 1(y - 3) + 1(z - 4) = 0$.
Step 4: Conclusion
$x - 2 + y - 3 + z - 4 = 0 \Rightarrow x + y + z - 9 = 0 \Rightarrow x + y + z = 9$. Final Answer: (A)
The normal to the plane is the vector joining the two points. The plane passes through the midpoint.
Step 2: Meaning
Midpoint $M = (\frac{1+3}{2}, \frac{2+4}{2}, \frac{3+5}{2}) = (2, 3, 4)$. Direction ratios of the normal $\vec{n} = (3-1, 4-2, 5-3) = (2, 2, 2)$, which is equivalent to $(1, 1, 1)$.
Step 3: Analysis
Equation of plane: $1(x - 2) + 1(y - 3) + 1(z - 4) = 0$.
Step 4: Conclusion
$x - 2 + y - 3 + z - 4 = 0 \Rightarrow x + y + z - 9 = 0 \Rightarrow x + y + z = 9$. Final Answer: (A)
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