Find the equation of a line which passes through the point (22, −6) and is such that the intercept of x-axis exceeds the intercept of y-axis by 5.

#### Solution

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

Here, a = b + 5 ... (1)

The line passes through (22, −6).

∴\[\frac{22}{a} - \frac{6}{b} = 1\] ... (2)

Substituting *a* = *b* + 5 from equation (1) in equation (2)

\[\frac{22}{b + 5} - \frac{6}{b} = 1\]

\[ \Rightarrow 22b - 6b - 30 = b^2 + 5b\]

\[ \Rightarrow b^2 - 11b + 30 = 0\]

\[ \Rightarrow \left( b - 5 \right)\left( b - 6 \right) = 0\]

\[ \Rightarrow b = 5, 6\]

From equation (1)

When b = 5 then, a = 5 + 5 = 10

When b = 6 then, a = 6 + 5 = 11

Thus, the equation of the required line is

\[\frac{x}{10} + \frac{y}{5} = 1 or \frac{x}{11} + \frac{y}{6} = 1\]

\[ \Rightarrow x + 2y - 10 = 0 \text { or }6x + 11y - 66 = 0\]