Lines parallel to the Axes

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Consider a line which is parallel to the x-axis, intersecting the y-axis at the point \(\left( {0,k} \right)\):

Line parallel to x-axis, intersecting y-axis

If \(P\left( {x,y} \right)\) 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 x-axis at the point \(\left( {k,0} \right)\) :

Line parallel to y-axis, intersecting x-axis

An 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:

Lines parallel to x and y axes forming rectangle

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