Question

In three-dimensional space, the equation $3x - 4y = 0$ represents.
$\left( a \right){\text{ A plane containing Z - axis}}$
$\left( b \right){\text{ A plane containing X - axis}}$
$\left( c \right){\text{ A plane containing Y - axis}}$
$\left( d \right){\text{ None of these}}$

Answer 1

So for solving such questions, as we know already that on putting any value of $z$ equation will have no effect as we knew that $0 \times z = 0$ , So by all this information and equating the value we will get the answer for such a type of question.

Complete step-by-step answer:
We have the equation given as $3x - 4y = 0$ and since it is in three-dimensional space and if we consider the $z - axis$ also then the equation will become,
$\Rightarrow 3x - 4y + 0z = 0$
So it means that while substituting any value of $z$ , then we can say that there will not be any effect on it. So mathematically it can be written as
$\Rightarrow 0 \times z = 0$
Therefore, we can say that it will be a plane containing the $z - axis$ and also it will pass through all points on $z$ if it satisfies the condition for $\left( {x,y} \right)$ .
Hence, the option $\left( a \right)$ is correct.

Note: Here, a point should be noted that the option $\left( d \right)$ is seen that it is not right because of its plane, not a line. So in this case it will pass through $\left( {0,0,0} \right)$ not the coordinate $\left( {0,0} \right)$ . So a plane requires three non-collinear points to be specified. And if there are only two points then there will be an infinite number of planes containing the axis. For each value, we will get a unique plane passing through it. Hence, to find the equation for a plane we need a point on the plane and a vector that will be orthogonal to the plane. These are the information we should know about solving a question like this.