Let P be the plane passing through the intersection of the planes vecrcdot (hati+3hatj-hatk )=5 and vecrcdot (2hati-hatj+hatk )=3 , and the point (2,1,-2) . Let the position vectors of the points X and Y be hati-2hatj+4hatk and 5hati-hatj+2hatk respectively. Then the points

3D Geometry: Let P be the plane passing through the intersection of the planes… | JEE Main 2022

JEE Main 2022 (Online) 25th June Evening ShiftQuestion 1

Let $P$ be the plane passing through the intersection of the planes $\vec{r}\cdot\left(\hat{i}+3\hat{j}-\hat{k}\right)=5$ and $\vec{r}\cdot\left(2\hat{i}-\hat{j}+\hat{k}\right)=3$, and the point $(2,1,-2)$. Let the position vectors of the points $X$ and $Y$ be $\hat{i}-2\hat{j}+4\hat{k}$ and $5\hat{i}-\hat{j}+2\hat{k}$ respectively. Then the points

3D Geometry

$X$ and $X+Y$ are on the same side of $P$$Y$ and $Y-X$ are on the opposite sides of $P$$X$ and $Y$ are on the opposite sides of $P$✓ Correct answer$X+Y$ and $X-Y$ are on the same side of $P$

Solution

Plane: $(x+3y-z-5)+k(2x-y+z-3)=0$; point $(2,1,-2)$: $2+k(-2)=0\Rightarrow k=1$, so $P:3x+2y-8=0$. $X=(1,-2,4)$: $3-4-8=-9<0$. $Y=(5,-1,2)$: $15-2-8=5>0$. $X+Y=(6,-3,6)$: $4>0$. $Y-X=(4,1,-2)$: $6>0$. $X-Y=(-4,-1,2)$: $-22<0$. X,Y opposite sides: (C) true; (A) false, (B) false, (D) false.

3D Geometry - JEE Mathematics question (JEE Main 2022 (Online) 25th June Evening Shift)

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