JEE 2022 — Mathematics PYQ

1. Condition for a line to lie on a plane:

If a line lies on a plane, any point on the line must satisfy the plane equation, and the line must be perpendicular to the normal of the plane.

• Point on the line: $(2,-1,-3)$

• Direction ratios of the line: $(3,-2,-1)$

• Normal vector to the plane: $(p,-q,1)$

2. Point substitution in the plane equation:

Substitute $(2,-1,-3)$ into $px-qy+z=5$:

3. Perpendicular condition:

The dot product of the line's direction and the plane's normal must be zero:

4. Solving for $p$ and $q$:

Multiply equation $(i)$ by 2:

Subtract equation $(ii)$ from this:

Substitute $p=15$ into equation $(i)$:

5. Equation of the Plane:

Substitute $p=15$ and $q=-22$ into the plane equation $px-qy+z=5$:

6. Shortest distance from the origin $(0,0,0)$:

The formula for distance $d=\frac{|a{x}_1+b{y}_1+c{z}_1+d|}{\sqrt{a^2+b^2+c^2}}$:

Final Answer: (B) $\sqrt{\frac{5}{142}}$