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Let plane Pi be parallel to the y-axis and contain the points (1, 0, 1) and (3, 2, -1). Points A = (4, 0, 0) and B = (6, 0, -2) are given. P = (x0, y0, z0) is a variable point on Pi. Match each item: (I) If Pi: x + a*y + b*z = c, find |a + b + c|. (II) If (PA + PB) is minimum, find |4*x0 + y0 + 2*z0|. (III) If (PA + PB) is minimum, find |x0 + y0 + z0|. (IV) If the reflection of line AB in Pi is (x-2)/1 = (y-alpha)/0 = (z+beta)/(-1), find (alpha⁴ + beta⁴). List-II: (P) 16, (Q) 12, (R) 3, (S) 2, (T) 10

1. I→R; II→Q; III→P; IV→S
2. I→Q; II→R; III→S; IV→P
3. I→R; II→Q; III→S; IV→P
4. I→R; II→S; III→Q; IV→P

Correct answer: I→R; II→Q; III→S; IV→P

Solution

Pi: x + z = 2 (a=0, b=1, c=2). (I) |0+1+2|=3=R. Reflect A=(4,0,0) across x+z=2 to get A'=(2,0,-2); line A'B meets Pi at P=(4,0,-2). (II) |4(4)+0+2(-2)|=12=Q. (III) |4+0-2|=2=S. Reflect A,B to find reflected line passes through (2,0,-2) with direction (1,0,-1); alpha=0, beta=2; alpha⁴+beta⁴=16=P. (IV)→P.

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