Q.364. Number of tangents to a curve and points of contacts.

Question 364.
How many tangent lines to the curve y= x/(x + 1) pass through the point (1,2)? At which points do these tangent lines touch the curve?

Answer 364.
The line through (1, 2) having slope m is
y - 2 = m (x - 1)

The tangent to a curve is a line which intersects the curve in two or more identical points. Hence, solving the above line with the curve,
y = x/(x + 1) = 2 + m(x - 1)
=> x = 2(x + 1) + m (x^2 - 1)
=> mx^2 + x + 2 - m = 0
The roots of this quadratic eqn. must be identical
=> its discriminant = 0 and the equal roots are x = 1/(2m)
=> 1 - 4m(2 - m) = 0
=> 4m^2 - 8m + 1 = 0
=> m^2 - 2m + 1/4 = 0
=> (m - 1)^2 = 3/4
=> m - 1 = ±√3/2
=> m = (1/2) (2 ± √3)
Two values of m indicate that there are two tangent lines.

The points of contact are given by
x = - 1/2m = - (2 - √3) or - (2 + √3)
For x = - (2 - √3),
y = x/(x+1) = - (2 - √3)/(-1 + √3) = - (1/2) (2 - √3)*(√3 + 1)   ...   (by rationalizing)
=> y = (1/2) (1 - √3)
Similarly, for x = - (2 + √3), y = (1/2)(1 + √3)
=> the points of intersection are
(-2 ± √3, (1 ∓ √3)/2).

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