The equation of a curve whose normal at any point has a slope which is the same as the ordinate and which passes through (1,-1) is 2 x= k (3- y ^2 ) . Then k is

Step-by-step solution

The slope of the normal at point (x, y) equals the ordinate y , establishing the differential equation - 1 dy dx = y . Rewriting gives dy dx = - 1 y , a separable equation. Separating variables and integrating yields y ,dy = -dx , so y^2 2 = -x + C . Using the initial condition (1, -1) : 1 2 = -1 + C , thus C = 3 2 . The curve becomes y^2 2 = -x + 3 2 , or 2x = 3 - y^2 after rearrangement. Comparing with the form 2x = k(3 - y^2) confirms k = 1 . Final Answer: A