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