Question

Let P be the plane passing through the intersection of the planes

r→.(i+3k−k)=5 and r→ .(2i−j+k)=3,

and the point (2, 1, –2). Let the position vectors of the points X and Y be

i−2j+4k and 5i−j+2k

respectively. Then the points

• A. X and X + Y are on the same side of P
• B. Y and Y – X are on the opposite sides of P
• C. X and Y are on the opposite sides of P
• D. X + Y and X – Y are on the same side of P

Answer 1

Answer: (A)

X and X + Y are on the same side of P

Answer 2

The correct option is(C): X and Y are on the opposite sides of P.

Let the equation of required plane

$\pi:(x+3y-z-5)+λ(2x-y+z-3)=0$

$∵(2,1,-2)\,\text{lies on it so,} 2+λ(-2)=0$

⇒λ=1

Hence,

$\pi:3x+2y-8=0$

$∵\pi{x}=-9,\pi{y},\pi_{x+y}=4$

$\pi_{x+y}=-22\,and\,\pi_{y-x}=6$

Clearly, X and Y are on opposite sides of plane π.