Intercept Form of a Straight Line Equation

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Lets discuss another form. Notice that to uniquely determine any straight line, we either need the slope of the line and a point through which this line passes, or we need at least two points through which that line passes. Thus for example, a line can also be uniquely determined if we are given the two points where this line intersects the x-axis and the y-axis.

Notice that  \(\begin{align}\tan \phi = \frac{b}{a}\end{align}\) so that the slope of the line is \(m = \tan \theta = \,\,\tan \left( {\pi - \phi } \right) = - \tan \phi = \begin{align}- \frac{b}{a}\end{align}.\) Also, the y-intercept is b. Thus, using the slope intercept form obtained earlier, the equation of the line L is

\[\begin{align}  &\qquad\quad y =  - \frac{b}{a}x + b \\& \Rightarrow \quad bx + ay = ab  \\&
   \Rightarrow \quad  \boxed{\,\frac{x}{a}\,\, + \,\,\frac{y}{b} = 1\,}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,:{\;\mathbf{Intercept}}{\text{ }}{\mathbf{form}}  \\&  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \nearrow \,\,\,\,\,\,\,\,\,\quad \nwarrow   \\& x - {\text{intercept}}\quad y - {\text{intercept}} \end{align} \]

Thus, if we know the x and y intercepts, we can directly use this form to write the equation of the line.

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