Question

If the tangent to the curve \(xy + ax + by = 0\) at \((1,1)\) makes an angle \(\tan^{-1} 2\) with X-axis, then find \(\frac{ab}{a+b}\).

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Use implicit differentiation and given slope condition to find relation between \(a,b\).

Answer 1

The Correct Option is B

Equation: \[ xy + ax + by = 0. \] Implicit differentiation at \((1,1)\): \[ y + x \frac{dy}{dx} + a + b \frac{dy}{dx} = 0. \] Slope at \((1,1)\): \[ \frac{dy}{dx} = -\frac{y + a}{x + b}. \] Given slope = \(\tan \theta = 2\). Substitute \(x=y=1\): \[ 2 = -\frac{1 + a}{1 + b} \implies 2(1+b) = -(1+a). \] Solve for \(a,b\), then calculate \(\frac{ab}{a+b} = 2\).