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Question
Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
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Solution
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
\[\frac{x}{3} + \frac{y}{6} = 1 \text { or }\frac{x}{6} + \frac{y}{3} = 1\]
\[ \Rightarrow 2x + y = 6 \text { or } x + 2y = 6\]
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