Question

For what values of c is the curve y = c/(x+1) tangent to the line through the points (0,3) and (5, -2)?

Answer 1

help me please :)

Answer 2

have you found the equation of the line through those two points?

Answer 3

is it y = -x + 3?
this is for the equation for the tangent line.
am I correct?

Answer 4

@UnkleRhaukus

Answer 5

yes that is the line through the two points

Answer 6

what should I do next?

Answer 7

@UnkleRhaukus

Answer 8

when do the equations y = c/(x+1)
y = -x + 3

meet?

Answer 9

i think you have to find the value for c which makes one solution to those equations

Answer 10

ohh, so I just need to equate those two and solve for c?
am I correct?

Answer 11

well equate them yes,

c/(x+1) = -x + 3

what so can simply it to

Answer 12

c = (-x + 3)(x+1)
is that it?

Answer 13

Why did my teacher got the derivative of y = c/(x+1)?
He didn't finish the solution, so I don't know if i'm right.

Answer 14

you want there to be exactly one solution,

you'll have to expand the brackets and complete the square,

Answer 15

oh, we can use derivatives?

Answer 16

Our topic is differential calculus , so I think I need to use derivatives here.
yes

Answer 17

oh right sorry, i was thinking of a different approach.

Answer 18

well can you take the derivative of
y = c/(x+1)

Answer 19

\[\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}\]

Answer 20

I got y' = -c/(x+1)^2

Answer 21

thats right

Answer 22

what should I do next?

Answer 23

This is were I'm stuck :)

Answer 24

i dont really understand this question

Answer 25

ohhh

Answer 26

I think the next step is

-c/(x+1)^2 = -1

because the derivative of y = c/(x+1) is it's tangent line

Answer 27

yeah, .,

Answer 28

How about this, using discriminants?
y = c / (x + 1)
y = -x + 3
-x + 3 = c / (x + 1)
(-x + 3)(x + 1) = c
-x^2 + 2x + 3 = c
x^2 - 2x - 3 = -c
x^2 - 2x + c - 3 = 0
b^2 - 4ac = 0
(-2)^2 - 4(1)(c - 3) = 0
4 - 4c + 12 = 0
4c = 16
c = 4

Answer 29

@TURITW Why is it that b^2 - 4ac = 0 ?

Answer 30

b^2 - 4ac > 0 means 2 real and distinct roots.
b^2 - 4ac = 0 means 2 real and equal roots.
b^2 - 4ac < 0 means 2 complex roots.

Answer 31

Why do you need to have equal roots?

Answer 32

Equal roots means tangent, because the curve needs to touch the line once

Answer 33

Why is it that equal roots means tangent?
I'm sorry if it's a dumb question. :)

Answer 34

@TURITW

Answer 35

Equal roots means same coordinates, so it means it only cuts the line once, example,
It cuts the line at (1 , 1) (just a random coordinates) only.

Answer 36

Thanks, I'll just show my teacher two different solutions.
I'll just ask for the correct answer. :)

Answer 37

Thanks to both of you. :)