Hey guys,
Here is another question I am struggling with:
Find the equation of line with the slope -1 that is tangent to the curve 1/(x-1).
(5 marks: 1 mark each for a graph, for the general equation of the line, for getting the quadratic, for solving for k, for the solution)
So far I have this. I am taught in my book to solve by setting the y-values equal to each other.
Hmmm, can you clarify what you did in the first line there? Why you changed all of the signs and got rid of an x and added the 1 to the k's? I could just figure out the question from your help, but I also like to know what happened so that I can actually learn it and not just replicate it. I understand the rest except for that part
I moved all terms to the LHS:
Then I factored out -x at the blue terms and collected the constants in one bracket (this step isn't necessary here, but I wanted to make clear this part)
Additional remark: Where the (k+1) comes from:
If you want to factor out a value from a sum you have to divide each summand by the factor. The result of the divison is put into bracket:
Alright, thank you very much.
However I can't find the right answer?
In the book with this kind of question, they end up with a nice easy answer. For example, from the formula y= x^2 + 5x +6 -x, they come up with k= -1/4.
When I proceed with what you left me off with. I come up with:
k^2-2k+5 = 0
Now, I am not sure if I did that right, because I was a little confused as to the reason you rearanged the formula in the last line you gave me. To me, it gave the same answer as if I worked it out in left side of that statement.
Anyways, I am wondering why I am left with another quadratic equation when I should have a single answer? Argh, why couldn't they make it easy like the example they give me?
I am supposed to use the quadratic formula again on that? But then k is no longer in my equation? And I am supposed to get a single answer, not roots.. So I am confused.
I re-arranged the last line to give you a hint how to solve the equation without much work:
This is a product of 2 factors which is only zero if one factor is zero too:
Now plug in this value to get the x-coordinate of the tangent point: (there are 2 tangent points!):
T_1(0, -1), and T_2(2, 1)