The points on the curve y ^2= x^3 9 , where the normal to the curve makes equal intercepts with the axes, are

Step-by-step solution

The differential equation dy dx = (x+y) + (x+y) can be solved using the substitution z = x+y . Differentiating gives dz dx = 1 + dy dx , so dy dx = dz dx - 1 . Substituting into the original equation yields: dz dx - 1 = z + z dz dx = 1 + z + z Separating variables: dz 1 + z + z = dx Using the tangent half-angle substitution t = ( z 2 ) , where dz = 2dt 1+t^2 , z = 1-t^2 1+t^2 , and z = 2t 1+t^2 : 2dt 1+t^2 1 + 1-t^2 1+t^2 + 2t 1+t^2 = dx Simplifying the denominator: (1+t^2) + (1-t^2) + 2t = 2 + 2t 2dt 2(1+t) = dx d