For what value of c is the curve y = c/(x + 1) tangent to the line through the points (0, 3) and (5, -2)?

Solution:

Given,

The curve y = c/(x + 1).

The points (0, 3) and (5, -2).

The value of c is required such that the line joining the points (0, 3) and (5, -2) is tangent to the curve y = c/(x + 1).

The slope of the line joining (0, 3) and (5, -2) is given by (y₂ - y₁)/ (x₂ - x₁)

Slope = 3 + 2/0 - 5 = -1.

This is equal to the slope of tangent on the curve and that is given by,

dy/dx = -c/(x + 1)²

dy/dx = -1

⇒ -c/(x + 1)² = -1

c = (x + 1)² ------> (1)

Equation of the line joining the points (0, 3) and (5, -2) is given by

(y - 3) = -1(x - 0)

y = -x +3

Solving the equation of tangent and the curve for the point of intersection

c/(x + 1) = -x+3

x + c/(x + 1) = 3---------> (2)

Substituting (1) in (2), we solve for x.

x + (x + 1)² /(x + 1) = 3

x +x + 1 = 3

x = 1

Putting x = 1 in (2), we get

c= 4.

Therefore, c = 4

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For what value of c is the curve y = c/(x + 1) tangent to the line through the points (0, 3) and (5, -2)?

Summary,

Value of c for the curve y = c/(x + 1) tangent to the line through the points (0, 3) and (5, -2) is 4.