Question

Find the equation of the straight line passing through $\left( { - 2,4} \right)$ and making non-zero intercepts whose sum is zero.

Answer 1

In this example, first we will write the equation of the straight line making non-zero intercepts. Then, we will use the given condition that the sum of intercepts is zero. Also given that the line is passing through the point $\left( { - 2,4} \right)$. So, we will put $x = - 2$ and $y = 4$ to find the required line.

Complete step-by-step solution:
We know that the equation of the straight line making non-zero intercepts $a$ and $b$ on $X$axis and $Y$axis respectively is given by $\dfrac{x}{a} + \dfrac{y}{b} = 1\; \cdots \cdots \left( 1 \right)$.
Here given that the sum of intercepts is zero. Therefore, $a + b = 0 \Rightarrow a = - b$.
Now we are going to put $a = - b$ in the equation $\left( 1 \right)$. Therefore,
$\- \dfrac{x}{b} + \dfrac{y}{b} = 1 \\\ \Rightarrow \dfrac{{y - x}}{b} = 1 \\\ \Rightarrow y - x = b\; \cdots \cdots \left( 2 \right) \\\$
Also given that the line is passing through the point $\left( { - 2,4} \right)$. Now we will put $x = - 2$ and $y = 4$ in the equation $\left( 2 \right)$. Therefore,
$4 - \left( { - 2} \right) = b \\\ \Rightarrow 4 + 2 = b \\\ \Rightarrow b = 6 \\\$
Now we will put the value of $b$ in the equation $\left( 2 \right)$ to find the required line. Therefore, we get
$y - x = 6$ which is the equation of the straight line passing through $\left( { - 2,4} \right)$ and making non-zero intercepts whose sum is zero.

Note: If we need to find the equation of the straight line making non-zero equal intercepts then we will use $\dfrac{x}{a} + \dfrac{y}{a} = 1$. That is, $x + y = a$. Also we can use $x + y = b$.