Question

Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.

Answer 1

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

Here, a + b = 9  
\[\Rightarrow b = 9 - a\] ... (1)
The line passes through (2, 2).

∴ \[\frac{2}{a} + \frac{2}{b} = 1\]  ... (2)
From equations (1) and (2)

\[\frac{2}{a} + \frac{2}{9 - a} = 1\]

\[ \Rightarrow 18 - 2a + 2a = 9a - a^2 \]

\[ \Rightarrow a^2 - 9a + 18 = 0\]

\[ \Rightarrow \left( a - 3 \right)\left( a - 6 \right) = 0\]

\[ \Rightarrow a = 3, 6\]

For a = 3, b = 9  \[-\] 3 = 6

For a = 6, b = 9  \[-\] 6 = 3

Thus, the equation of the line is

\[\frac{x}{3} + \frac{y}{6} = 1 \text { or }\frac{x}{6} + \frac{y}{3} = 1\]

\[ \Rightarrow 2x + y = 6 \text { or } x + 2y = 6\]