The slope of the tangent line to a curve at any point (x, y) on the curve is x/y. What is the equation of the curve if (3, 1) is a point on the curve?

Question

A. x^2 + y^2 =8 <-------My choice
B. x+y=8
C. x^2 - y^2=8
D. xy=8

Is my solution correct? Thank you :)

Answer 1

Not quite. Your choice has dy/dx = -x/y
Also, the point does not lie on your curve.

Answer 2

ahhhh, ok. So it's x^2 - y^2=8, right? dy/dx of that choice gives me x/y. Because I got that, does that necessarily mean that it is the right answer?

Answer 3

I'd say yes, maybe depending on how you arrived at that answer. I assume you actually worked things out.

dy/dx = x/y
y dy = x dx
1/2 y^2 = 1/2 x^2 + C
or,
x^2 - y^2 = C
now plug in your point (3,1) to find that C=8