Question

Find the values of a and b for the curve xy^2 + ay = bx such that at the point (1,-1), the curve has a slope of -1/2.

Answer 1

You're familiar with derivatives, correct?

Answer 2

Yes

Answer 3

Well, we need to find the derivative of this function and then we need to go and plug in the in the value of x,y and set that= -1/2

a and b are considered to be constants in this case so can you differentiate this equation with respect to x and tell me what you get?

Answer 4

\[\frac{ dy }{ dx }= \frac{ b - y^2 }{ 2xy+a }\]

Answer 5

OK, we have the derivative, correct.

I'm trying to find a system of solutions to help find a and b,

Answer 6

\[\frac{ b-y^2 }{ 2xy+a } = -\frac{ 1 }{ 2 }\]

Answer 7

That's one equation. We need another one

Answer 8

since y = -1 and x = 1. Shouldn't I switch the x's and y's to these numbers?

Answer 9

Yes.

Answer 10

You mean plug in 1 for x and -1 for y?

Answer 11

Yes

Answer 12

\[xy^2 + ay = bx\]

Answer 13

(1)(-1)^2 + a(-1) = b(1)
1 - a = b

Answer 14

I'm not sure if that equation would work though. We can try

Answer 15

the given point must satisfy :
f(x) = 0
f'(x) = -1/2

Answer 16

Plug in (1, -1) both in the equation of the curve and in the equation of the derivative.
That will give you two equations in a and b.

Answer 17

so now I just solve the derivative of the function

Answer 18

Yes what @mathstudent55 confidently corrected my thought lol

Answer 19

\[\frac{ (1-a) - (-1)^2 }{ 2(1)(-1)+a } = -\frac{ 1 }{ 2 }\]

Answer 20

Ok and that simplifies?

Answer 21

a = -2

Answer 22

correct

Answer 23

oo I just plug in -2 to the 1 - a = b to find b

Answer 24

1 - (-2) = b
1 + 2 = b
b = 3
everyone left immediately =.= thanks though

Answer 25

Correct!

Answer 26

Excellent work, @Mateaus !

Answer 27

Thanks everyone.

Answer 28

You're welcome.