# Lines parallel to the Axes

Consider a line which is parallel to the ** x-axis**, intersecting the

**at the point \(\left( {0,k} \right)\):**

*y*-axisIf \(P\left( {x,y} \right)\) is any arbitrary point on this line, the *y*-coordinate of *P* must be equal to *k* (since the line is horizontal – all points have the same *y*-coordinate). Thus, we can say that the coordinates of *P* are constrained by the equation

\[y = k\]

In other words, the equation of the line in the figure above is \(y = k\). Note that there is no *x* in the equation because *y* equals *k* regardless of the value of *x*.

Now, consider a line which is parallel to the ** y-axis**, intersecting the

**at the point \(\left( {k,0} \right)\) :**

*x*-axisAn arbitrary point \(P\left( {x,y} \right)\) on this line will be constrained by the equation

\[x = k\]

Thus, the equation of this line is \(x = k\).

**Example 1:** Find the area of the rectangle formed by the lines \(x = 3\), \(x = - 2\), \(y = 1\) and \(y = - 3\).

**Solution:** The figure below shows the four lines, and the rectangle enclosed:

<

Clearly, the area of the rectangle is \(5 \times 4 = 20\) square units.