3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
The equation of the plane parallel to plane $x + y + 2z = 5$ at a distance $\sqrt{6}$ units from the plane is/are:
x + y + 2z + 11 = 0
x + y + 2z = 11
x + y + 2z + 1 = 0
x + y + 2z - 1 = 0
Step-by-Step Solution
Key Concept: Use the distance formula between parallel planes: $d = rac{|c_1 - c_2|}{\sqrt{a^2 + b^2 + c^2}}$ where planes are $ax + by + cz = c_1$ and $ax + by + cz = c_2$.
A plane parallel to $x + y + 2z = 5$ has the form $x + y + 2z = k$ for some constant $k$. The distance between parallel planes $x + y + 2z = 5$ and $x + y + 2z = k$ is given by $d = rac{|k - 5|}{\sqrt{1^2 + 1^2 + 2^2}} = rac{|k - 5|}{\sqrt{6}}$. Setting this equal to $\sqrt{6}$: $ rac{|k - 5|}{\sqrt{6}} = \sqrt{6}$ gives $|k - 5| = 6$, so $k = 11$ or $k = -1$. The two planes are $x + y + 2z = 11$ and $x + y + 2z = -1$ (or equivalently $x + y + 2z + 1 = 0$).
Correct Answer: 2,3