If the tangent to the curve \(xy+ax+by = 0\) at \((1,1)\) makes an angle of \(tan^{-1}2\) with positive direction of the \(x\)-axis, then the value of \(\frac{ab}{a+b}\) is...
Show Hint
Use the point to get one equation and the slope tan(angle) = 2 to get another.
Step 1: Two equations:
(i) $a + b = -1$ from the point. (ii) $-\frac{1 + a}{1 + b} = 2$ from the slope.
Step 2: Solve:
Put $a = -1 - b$ into (ii): $-\frac{-b}{1 + b} = 2$, so $b = 2 + 2b$, giving $b = -2$ and $a = 1$.
Then $ab = -2$ and $a + b = -1$, so the ratio is $2$.
Final Answer:
The value is $2$, option (C).
\[ \boxed{2} \]