Question

Given a plane E and a line I contained in E, so that two Half planes ${{H}_{1}}$ and ${{H}_{2}}$ are formed, then line I lies
(a) in ${{H}_{1}}$ only
(b) in ${{H}_{2}}$ only
(c) in both ${{H}_{1}}$ and ${{H}_{2}}$
(d) neither in ${{H}_{1}}$ nor in ${{H}_{2}}$

Answer 1

Hint : For solving this question you should know about a plane and a line which is subjected to the plane. In this problem first we will discuss a plane and the line. And this line is subjected to the plane which means if anyone changes them it will surely affect others too.

Complete step-by-step solution:
According to our question it is asked that given a plane E and a line I, contained in E, so that two Half planes ${{H}_{1}}$ and ${{H}_{2}}$ are formed then line I lies in which planes.
As we know that if any plane is given to us then that means it is a complete plane and it can be an infinite plane or a finite plane. And if any line is given to us then it must be finite.
And if it is given that the line is subjected with or the plane is subjected with line which means both are subjected with each other then it is confirmed that the plane is also finite plane and both are finite and if anyone changes to itself then both will change with respect to each other.
Here, if we look at the question then the plane is given as E. And the line is given as I.
And the line is dividing the plane E in two half planes ${{H}_{1}}$ and ${{H}_{2}}$.
So, the line I will lie in both ${{H}_{1}}$ and ${{H}_{2}}$ because this is the subject of the plane and it divides that.

So, the correct option is (c).

Note: While solving those types of questions you have to be careful about the statements of the question. If the equation is given of a line and a plane and if these satisfy each other then these will be provided the unknown values of each other. And this is done by comparing or using both equations with each other.