Q.$x=a$ represent a plane parallel to ________.

The equation $x = a$ describes the set of all points in three-dimensional space where the $x$-coordinate is fixed at the constant value $a$, while the $y$ and $z$ coordinates can take any real value.

Since $y$ and $z$ are free to vary independently, every point of the form $(a, y, z)$ lies on this surface. This means the plane extends infinitely in both the $y$-direction and the $z$-direction. …

Understanding planes in 3D coordinate geometry

When we write an equation like $x = a$ in three-dimensional space, we're imposing a constraint on only one coordinate while leaving the other two completely free. This freedom is what creates a plane rather than a line or point.

Think about what $x = a$ actually means: every point on this surface has the same $x$-coordinate (namely $a$), but $y$ and $z$ can take any value whatsoever. So the point $(a, 0, 0)$ is on this plane, as is $(a, 5, -3)$, $(a, 100, -200)$, and infinitely many others.

Step-by-step reasoning

1. Identify what varies and what's fixed

   The equation $x = a$ fixes the $x$-coordinate at the value $a$, while $y$ and $z$ are unrestricted. We can write this as the set of all points $(a, y, z)$ where $y, z \in \mathbb{R}$.

2. Recognize the geometric shape

   Since two coordinates are free to vary independently, this describes a two-dimensional surface—a plane. The plane "stands" at position $x = a$ along the $x$-axis.

3. Determine the orientation

   Because $y$ and $z$ span the entire plane while $x$ remains constant, this plane is perpendicular to the $x$-axis. Imagine slicing through 3D space with a knife held perpendicular to the $x$-axis at position $a$.

4. Find what it's parallel to

   The coordinate planes in 3D space are:

   • The $xy$-plane: where $z = 0$
   • The $yz$-plane: where $x = 0$ …