Finding all tangent lines through a point

[hlin818]

Homework Statement

Find all tangent lines of the graph f(x)=x+3/x that have a y intercept of 4.

Homework Equations

The Attempt at a Solution

Assume a is the x coordinate of a point of tangency. Thus the point of tangency is (a, a+3/a). We know the tangent line must pass through (0,4) so the slope of the line must be (a+3/a-4)/(a-0).

f'(x)=1-3/(x^2)

Derivative of f at point a must equal the slope of the tangent line, i.e. we must have
f'(a)=1-3/(a^2)=(a+3/a-4)/(a-0)=m

Solving I get a=3/2. However, looking at the graph of f(x), it seems there should be two points where the tangent line passes through 4, the other one being on the part of the f(x) where x<0. Where did I go wrong?

 

[Dick]

Homework Statement

Find all tangent lines of the graph f(x)=x+3/x that have a y intercept of 4.

Homework Equations

The Attempt at a Solution

Assume a is the x coordinate of a point of tangency. Thus the point of tangency is (a, a+3/a). We know the tangent line must pass through (0,4) so the slope of the line must be (a+3/a-4)/(a-0).

f'(x)=1-3/(x^2)

Derivative of f at point a must equal the slope of the tangent line, i.e. we must have
f'(a)=1-3/(a^2)=(a+3/a-4)/(a-0)=m

Solving I get a=3/2. However, looking at the graph of f(x), it seems there should be two points where the tangent line passes through 4, the other one being on the part of the f(x) where x<0. Where did I go wrong?

I don't think you went wrong anywhere. Looking at the graph I don't see any points x<0 where the tangent line will go through (0,4).

 

[ehild]

If a is negative, the y intercepts of the tangent lines are also negative.

ehild