Finding the values of K and the coordinates of the points of tangency.

Question

The line of 2x + 3y -1 = 0 is tangent to the graph y= 2/x +k. Find the exact values of k and the coordinates of the points of tangency

Answer 1

Slope of tangent line = -2/3

Find value of x that makes y' equal to -2/3

y' = -2/x^2 = 1/3

x = sqrt(3)

Use the equation of the tangent line to find the y coordinate of the point of tangency, given that x = sqrt(3):

2(sqrt3) + 3y - 1 = 0

y = (1 - 2sqrt(3)) / 3

Use the point of tangency coordinates to solve for k:

(1 - 2sqrt(3)) / 3 = 2 / sqrt(3) + k

This yields k = (1 - 2sqrt(3))/3 - 2/sqrt(3)