# Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1

## Homework Statement

Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1

y=1/x-1

## The Attempt at a Solution

Slope of -1 means y=-1x+k

So...
-1x+k = 1/x-1

I don't know how to rearrange this into a quadratic equation so that I can solve this.
I know I use b^2-4ac=0 and substituting for a, b and c.... I'm just stuck on rearranging the equation into the form: ax^2+bx+c=0

Mark44
Mentor

## Homework Statement

Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1

y=1/x-1

## The Attempt at a Solution

Slope of -1 means y=-1x+k

So...
-1x+k = 1/x-1

I don't know how to rearrange this into a quadratic equation so that I can solve this.
I know I use b^2-4ac=0 and substituting for a, b and c.... I'm just stuck on rearranging the equation into the form: ax^2+bx+c=0
You're missing a very important piece: how to find the slope of the tangent line to y = 1/x - 1.

I'm going to guess that you are in a class that has discussed how to find the tangent to a curve.

Mark44
Mentor
OTOH, maybe this actually is a precalculus-type problem. What does it mean to say that a line is tangent to a curve?

For your equation, -x + k = 1/x - 1, what about multiplying both sides by x?

OTOH, maybe this actually is a precalculus-type problem. What does it mean to say that a line is tangent to a curve?

For your equation, -x + k = 1/x - 1, what about multiplying both sides by x?

(-x+k)(x) = -x^2+k(x)
(1/x-1)(x) = ?

You're right, but I'm confused as to how to multiply the second part? x/x^2-x ?

Here's my second stab at it...

-1x+k = 1/x-1

Rearranged: -x^2+kx-1

Using the discriminant of the quadratic formula:

(b^2-4ac)

a=-1 b=k^2 c=-1

k^2-4(-1)(-1)=0
Solving:
k = -2

Therefore the equation is y=-1x-2
For the line with a slope of -1 that is tangent to the curve of 1/x-1
Can anybody verify if I got this right?

Mark44
Mentor
Here's my second stab at it...

-1x+k = 1/x-1

Rearranged: -x^2+kx-1
Where did the = go? Also, you have an error.
Using the discriminant of the quadratic formula:

(b^2-4ac)

a=-1 b=k^2 c=-1

k^2-4(-1)(-1)
Solving:
k = -2

Therefore the equation is y=-1x-2
For the line with a slope of -1 that is tangent to the curve of 1/x-1
Can anybody verify if I got this right?

Mark44
Mentor
(-x+k)(x) = -x^2+k(x)
(1/x-1)(x) = ?

You're right, but I'm confused as to how to multiply the second part? x/x^2-x ?
What is x * (1/x)?
What is x * (-1)?

Mark44
Mentor
From post #3:
What does it mean to say that a line is tangent to a curve?

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1

y=1/x-1

## The Attempt at a Solution

Slope of -1 means y=-1x+k

So...
-1x+k = 1/x-1

I don't know how to rearrange this into a quadratic equation so that I can solve this.
I know I use b^2-4ac=0 and substituting for a, b and c.... I'm just stuck on rearranging the equation into the form: ax^2+bx+c=0

What you wrote means $(1/x) - 1.$ Is that what you meant, or did you really mean $1/(x-1)$? If you meant the latter, you need to use brackets, but if you meant the former then what you wrote is OK.

RGV

Mark44
Mentor
What you wrote means $(1/x) - 1.$ Is that what you meant, or did you really mean $1/(x-1)$? If you meant the latter, you need to use brackets, but if you meant the former then what you wrote is OK.
I think he meant (1/x) - 1, but I'm not certain of it.