Question

Find the equations of the straight lines each of which passes through the point (3, 2) and cuts off intercepts a and b respectively on X and Y-axes such that a − b = 2.

Answer 1

The equation of the line with intercepts a and b is $\frac{x}{a}+\frac{y}{b}=1$

Here, a − b = 2
$\Rightarrow$a = b + 2 ... (1)

The line passes through (3, 2).

∴ $\frac{3}{a}+\frac{2}{b}=1$ ... (2)

Substituting a = b + 2 in equation (2)

$$\begin{gathered} \frac{3}{b+2}+\frac{2}{b}=1 \\ \Rightarrow 3b+2b+4=b^2+2b \\ \Rightarrow b^2-3b-4=0 \\ \Rightarrow \left(b-4\right)\left(b+1\right)=0 \\ \Rightarrow b=4,-1 \end{gathered}$$

Now, from equation (1)

For b = 4, a = 4 + 2 = 6

For b = − 1, a = − 1 + 2 = 1

Thus, the equations of the lines are

$$\begin{gathered} \frac{x}{1}+\frac{y}{-1}=1\mathrm{and}\frac{x}{6}+\frac{y}{4}=1 \\ \Rightarrow x-y=1\mathrm{and}2x+3y=12 \end{gathered}$$