SOLUTION: Hi; I need help with this multiple choice question. For the line y=x to be tangent to the curve with equation y = k / x

SOLUTION: Hi; I need help with this multiple choice question. For the line y=x to be tangent to the curve with equation y = k / x - 1

Question 1199827: Hi;
I need help with this multiple choice question.
For the line y=x to be tangent to the curve with equation y = k / x - 1
(k divided by x-1)
A. k=1
B. k>1
C. k>-1/4
D. k=1/4
E. 4k+1<0
Thanks.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

I'm assuming all of the "x-1" is in the denominator, and that you intend to say

If so, then you should wrap "x-1" in parenthesis and write y = k/(x-1)

I'll be using differential calculus to solve this problem.
If you aren't familiar with the topic, then please let me know.

Let's apply the derivative with respect to x.
Use the power rule where y = x^n leads to dy/dx = n*x^(n-1)

In this specific case:
y = k/(x - 1)
y = k(x - 1)^(-1)
dy/dx = -k(x - 1)^(-2)

The slope of the tangent is found through the derivative dy/dx
The tangent y = x has a slope of 1, which means dy/dx = 1
Use this fact to determine k in terms of x.

dy/dx = -k(x - 1)^(-2)
1 = -k(x - 1)^(-2)
(x - 1)^2 = -k
k = -(x-1)^2
This will be useful later.
Let's call this equation (1)

For the line y = x to be tangent to y = k/(x-1), the two must intersect.

Apply substitution.
y = k/(x - 1)
x = k/(x - 1)
x(x-1) = k
x(x-1) = -(x-1)^2 ........ refer to equation (1)
x^2-x = -(x^2-2x+1)
x^2-x = -x^2+2x-1
x^2-x+x^2-2x+1 = 0
2x^2-3x+1 = 0

Use the quadratic formula to solve for x.
I'll skip this subsection of steps.
You should get x = 1 and x = 1/2 = 0.5

If x = 1, then,
k = -(x-1)^2
k = -(1-1)^2
k = 0

If x = 1/2, then,
k = -(x-1)^2
k = -(1/2-1)^2
k = -1/4

We have either k = 0 or k = -1/4

If k = 0 happens, then
y = k/(x - 1)
becomes
y = 0/(x - 1)
y = 0
this line is completely flat and horizontal.
It overlaps the x axis.
Clearly y = x is not tangent to y = 0.
Therefore, we must rule out k = 0.

On the other hand if k = -1/4 = -0.25, then,
y = k/(x - 1)
y = -0.25/(x - 1)
y = (-1/4)/(x-1)
y = -1/(4(x - 1))
y = -1/(4x - 4)

I'll leave it to the student to verify the claim that y = x is tangent to y = -1/(4x - 4)
As another homework exercise, the student should try to find the point of intersection.

Graph:

Graphing tools such as Desmos and GeoGebra are very useful as quick visual verification methods.

Answer: k = -1/4

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