3D Geometry
Plane through line of intersection; distance from a point
nta_pyq_2023_jan
Grade 12
Question:
Let the equation of the plane passing through the line $x - 2y - z - 5 = 0 = x + y + 3z - 5$ and parallel to the line $x + y + 2z - 7 = 0 = 2x + 3y + z - 2$ be $ax + by + cz = 65$. Then the distance of the point $(a, b, c)$ from the plane $2x + 2y - z + 16 = 0$ is ______.
Step-by-Step Solution
Key Concept: Write plane as $P_1 + bP_2 = 0$, impose perpendicularity with direction of second pair to find $b$, then read off normal.
Plane $13x+10y+35z=65$, so $(a,b,c)=(13,10,35)$. Distance from $2x+2y-z+16=0$: $d = \frac{|26+20-35+16|}{3} = \frac{27}{3} = 9$. Answer: 9
Correct Answer: 9