Intercepts Form of a Line

If a and b are non-zero X and Y intercepts of a line l, then its equation is of the form

\frac{x}{a}  + \frac{y}{b} = 1


Since a is a X-intercept of the line l, as we know that any point lies on the X-axis its value of Y is equal to zero, so it passes through the point A\left( {a,0} \right). Also if b is the Y-intercept of the line l, and we know that any point lies on the Y-axis its value of X is equal to zero, so it passes through the point B\left( {0,b} \right) as shown in the given diagram.


inercepts-form-line

Now to prove intercepts form of a line, use the formula for two points form of a straight line as given by

\frac{{y  - {y_1}}}{{{y_2} - {y_1}}} = \frac{{x - {x_1}}}{{{x_2} - {x_1}}}


Take A\left( {a,0} \right) = \left( {{x_1},{y_1}}  \right) and B\left( {0,b} \right) =  \left( {{x_2},{y_2}} \right), putting these values in the above formula as

\begin{gathered} \Rightarrow \frac{{y - 0}}{{b - 0}} =  \frac{{x - a}}{{0 - a}} \\ \Rightarrow \frac{y}{b} = \frac{x}{{ - a}} -  \frac{a}{{ - a}} \\ \end{gathered}


  \Rightarrow \boxed{\frac{x}{a} + \frac{y}{b} = 1}


Which is the required equation of straight line in intercepts form.

Example: Find the equation of straight with X-intercept A\left( {3,0} \right) and Y-intercept B\left( {0,2} \right).
From the above information we have X-intercept is a = 3 and Y-intercept is b = 2, now putting all these values in the formula of intercepts form as given

\begin{gathered} \frac{x}{a} + \frac{y}{b} = 1 \\ \Rightarrow \frac{x}{3} + \frac{y}{2} = 1 \\ \Rightarrow 2x + 3y = 6 \\ \Rightarrow 2x + 3y - 6 = 0 \\ \end{gathered}


Which is the required equation of straight line.

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