Answer :

Slope of tangent is given as


Slope of tangent of a curve y = f(x) is given by




Integrate



Using partial fraction for




Equating numerator


A(x + 1) + Bx = 1


Put x = 0


A = 1


Put x = -1


B = -1


Hence


Hence equation (a) becomes




log(y – 1) = logx – log(x + 1) + c …(b)


Now it is given that the curve is passing through (1, 0)


Hence (1, 0) will satisfy the curve equation (b)


Putting values x = 1 and y = 0 in (b)


If we put y = 0 in (b) we get log (-1) which is not defined hence we must simplify further equation (b)


log (y – 1) – log x = – log (x + 1) + c


Using log a – log b = log




Using log a + log b = log ab



Take the constant c as log c so that we don’t have any undefined terms in our equation (Why only log c and not any other term because taking log c completely eliminates the log terms so we don’t have to worry about the undefined terms appearing in our equation)



Eliminating log



Now substitute x = 1 and y = 0



c = -2


Put back c = -2 in (c)



Hence the equation of curve is (y – 1)(x + 1) = -2x


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