If the tangent to the curve x y+a x+b y=0 at (1,1) is inclined at an angle ⁻¹ 2 with X -axis, then

Step-by-step solution

The given curve is xy + ax + by = 0 . Differentiating with respect to x , we get y + x dy dx + a + b dy dx = 0 . Rearranging for dy dx , we have dy dx (x + b) = -(y + a) , which implies dy dx = - y + a x + b . At the point (1, 1) , the slope of the tangent is m = - 1 + a 1 + b . The tangent is inclined at an angle ⁻¹ 2 with the X -axis, so m = ( ⁻¹ 2) = 2 . Equating the two expressions for m , we get - 1 + a 1 + b = 2 , which simplifies to -(1 + a) = 2(1 + b) , or 1 + a = -2 - 2b , which gives a + 2b = -3 . Since t