If the line joining the points `(0,3) and (5,-2)` is a tangent to the curve `y=C/(x+1)` , then the value of `C` is (a) 1 (b) `-2` (c) 4 (d) none of these

Options

A. 1
B. -2
C. 4
D. none of these

Answer

C

Explanation

The equation of the line joining (0,3) and (5,-2) is
` x+y-3=0 " "…(i) `
Suppose this line touches the curve `y=(C )/(x+1) " at " (alpha, beta ).` Then,
` ((dy)/(dx))_((alpha"," beta)) =` Slope of the line (i)
` rArr -(C)/((alpha+1)^(2))=-1 " "[ because (dy)/(dx)=- (C)/((x+1)^(2))]`
` rArr C=(alpha+1)^(2) " "...(ii) `
Also, ` (alpha,beta)` lies on the line (i) and the given curve. Therefore,
` alpha+ beta -3=0 " "...(iii) `
and, ` beta=(C)/(alpha+1) " "...(iv) `
Form (ii) and (iv), we get
` beta=alpha+1 " "...(v) `
Solving (iii) and (v), we get
` 2 alpha +1-3 =0 rArr alpha =1 `
Substituting ` alpha=1 ` in (ii), we get C = 4