Question

The value of \( c \) such that the straight line joining the points \[ (0,3) \quad {and} \quad (5,-2) \] is tangent to the curve \[ y = \frac{c}{x+1} \] is:

• A. \( 3 \)
• B. \( 4 \)
• C. \( 5 \)
• D. \( 2 \)

Hint:

For tangency conditions, equate the derivative of the function to the slope of the given line.

Answer 1

The Correct Option is B

Step 1: Finding the equation of the line The equation of the line through \( (0,3) \) and \( (5,-2) \): \[ y - 3 = \frac{-2 - 3}{5 - 0} (x - 0). \] Simplifying, \[ y = -x + 3. \] Step 2: Condition for tangency The given curve is \( y = \frac{c}{x+1} \). For tangency, we equate slopes: \[ \frac{d}{dx} \left( \frac{c}{x+1} \right) = -1. \] Solving for \( c \), we get: \[ c = 4. \]