Half Planes Made by a Straight Line

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\(\textbf{Art 6 : } \qquad\boxed{{\text{Half-planes}}}\)

Any straight line divides the Euclidean plane into two half planes. In this article, we wish to determine the half-plane in which an arbitrary point lies with respect to a given line.

Let the equation of the given line be  \(ax+by+c=0.\) Consider two points \(\left( {{x}_{1}},{{y}_{1}} \right)\,\,\text{and}\,\,\left( {{x}_{2}},{{y}_{ 2}} \right)\)that lie in different half-planes with respect to this line:

The point  \(\left( {{x}_{1}},{{y}_{1}} \right)\) lies in the lower half-plane while \(\left( {{x}_{2}},{{y}_{2}} \right)\) lies in the upper half plane. We require a condition on these co-ordinates which must be satisfied if the points lie in opposite half-planes. In Fig - 18, we have dropped verticle line segments from \(\left( {{x}_{1}},{{y}_{1}} \right)\,\,\text{and}\,\left( {{x}_{2}}{{y}_{2}} \right)\) to the given line, intersecting it in P and Q respectively.

The co-ordinates of P and Q are \(({{x}_{1}},y_{1}^{'})\) and  \(({{x}_{2}},y_{2}^{'})\) respectively where \(y_{1}^{'}\ne {{y}_{1}}\,\,\text{and}\,\,y_{2}^{'}\ne {{y}_{2}}\) .

Since P, Q lie on the given line, their co-ordinates must satisfy the equation of the line. Thus,

\[\begin{align}& \qquad\;\; a{{x}_{1}}+by_{1}^{\mathbf{'}}+c=0\qquad \Rightarrow \qquad y_{1}^{\mathbf{'}}=-\frac{\left( a{{x}_{1}}+c \right)}{b} \\ & and\quad a{{x}_{2}}+by_{2}^{\mathbf{'}}+c=0\qquad \Rightarrow \qquad y_{2}^{\mathbf{'}}=-\frac{\left( a{{x}_{2}}+c \right)}{b} \end{align}\]

Now, from Fig - 18 we have

\[\begin{align}&\qquad \qquad \qquad\quad\; {y_1} < y_1^{\mathbf{'}}\quad\text{and}\quad{y_2} > y_2^{\mathbf{'}}  \\&
   \Rightarrow  \;\;\quad {y_1} <  - \frac{{\left( {a{x_1} + c} \right)}}{b}\quad \text{and} \quad{y_2} >  - \frac{{\left( {a{x_2} + c} \right)}}{b}  \\&
   \Rightarrow  \quad \frac{{a{x_1} + b{y_1} + c}}{b} < 0\quad \text{and} \quad\frac{{a{x_2} + b{y_2} + c}}{b} > 0  \\&
 \Rightarrow  \qquad\quad \;  a{x_1} + b{y_1} + c\quad \text{and} \quad a{x_2} + b{y_2} + c\,\,\,{\text{are of opposite signs}}{\text{.}}  \\ 
\end{align} \]

This is the required condition. Translated into words, it says that for two points lying in opposite half-planes, their co-ordinates when substituted respectively into the equation of the line must give expressions of opposite signs. (For two points in the same half-plane, the signs would be the same).

As a corollary, observe that a point (x1, y1) lies in the same half–plane or opposite half–plane in which the origin lies accordingly as \(\left( {a{x_1} + b{y_1} + c} \right)\) and c are of the same sign or opposite signs respectively.

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