Q. 305.0( 1 Vote )
Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve any point (x, y) is 
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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