The 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

Since $a$ is an $X$-intercept of the line $l$, and as we know that if any point lies on the $X$-axis its value of $Y$ is equal to zero, 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 that lies on the $Y$-axis has a value of $X$ equal to zero, it passes through the point $B\left( {0,b} \right)$ as shown in the given diagram. Now to prove the intercepts form of a line, use the formula for the two points form of a straight line as given by

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

This is the required equation of a straight line in intercepts form.

Example: Find the equation of a straight line with $X$-intercept $A\left( {3,0} \right)$ and $Y$-intercept $B\left( {0,2} \right)$.

From the above information we know the $X$-intercept is $a = 3$ and the $Y$-intercept is $b = 2$. Now we put all these values in the formula of the intercepts form as given:

This is the required equation of a straight line.