EQUATION OF A LINE IN INTERCEPT FORM

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The equation of a line which cuts off the given intercept on both the coordinate axes is determined as follows:

Let PQ be a line meeting x-axis in A and y-axis in B.

Consider OA = a, and OB = b.

Then, the coordinates of A and B is\left( {a,0} \right){\rm{ and }}\left( {0,b} \right).

The equation of line joining A and B is written as follows:

\begin{array}{c}
y - 0 = \frac{{b - 0}}{{0 - a}}\left( {x - a} \right)\\
y =  - \frac{b}{a}\left( {x - a} \right)\\
\frac{y}{b} =  - \frac{x}{a} + 1\\
\frac{x}{a} + \frac{y}{b} = 1
\end{array}

Therefore, the equation a straight line in a intercept form is written as

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

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Example

The equation of a line which cuts off the intercepts 5 and -3 on x and y axes.

The general equation of straight line in intercept form is

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

Given that a = 5 and b = – 3

Substitute the value as follows:

\begin{array}{c}
\frac{x}{5} + \frac{y}{{ - 3}} = 1\\
\frac{{3x - 5y}}{{15}} = 1\\
3x - 5y = 15\\
3x - 5y - 15 = 0
\end{array}

Therefore, the required equation is{3x - 5y - 15 = 0}.

Question 1

Find the equation of a line which cuts off the intercepts 10 and 2 on x and y axes.

Solution 1

The general equation of straight line in intercept form is

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

Given, a = 10 and b = 2

Substitute the value as follows:

\begin{array}{c}
\frac{x}{{10}} + \frac{y}{2} = 1\\
\frac{{x + 5y}}{{10}} = 1\\
x + 5y = 10
\end{array}

Therefore, the required equation isx + 5y - 10 = 0.

 

Question 2

Find the intercepts made by the line 3x - 2y + 12 = 0on the coordinate axes.

Solution 2

The general equation of straight line in intercept form is

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

Consider the line3x - 2y + 12 = 0 and rewrite the equation as follows:

3x - 2y =  - 12

Divide both the sides of the equation by –12 as follows:

\begin{array}{c}
\frac{{3x}}{{ - 12}} - \frac{{2y}}{{ - 12}} = \frac{{ - 12}}{{ - 12}}\\
\frac{x}{{ - 4}} + \frac{y}{6} = 1
\end{array}

Therefore, the intercept on the axes are –4 and 6 respectively.

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