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?
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 :)
Not quite. Your choice has dy/dx = -x/y
Also, the point does not lie on your curve.
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
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?
To find the equation of the curve, we can use calculus to represent the slope of the tangent line at any given point. The given information states that the slope of the tangent line at any point (x, y) on the curve is x/y.
To proceed, we can take the derivative of the equation y = f(x) with respect to x, denoted as dy/dx. This derivative represents the slope of the tangent line at any point.
dy/dx = x/y
Now, we can solve the equation to find the function f(x). Rearranging the equation, we get:
y dy = x dx
Integrating both sides:
∫y dy = ∫x dx
(1/2) y^2 = (1/2) x^2 + C
where C is the constant of integration.
Now, to find the equation of the curve, we can substitute the given point (3, 1) into the equation and solve for C.
(1/2) (1)^2 = (1/2) (3)^2 + C
1/2 = 9/2 + C
-4 = C
Replacing C in the previous equation:
(1/2) y^2 = (1/2) x^2 - 4
Multiplying both sides by 2 to clear the denominators:
y^2 = x^2 - 8
Therefore, the equation of the curve is x^2 - y^2 = 8.
However, from the options provided, it seems that there is a slight typo in option A. The correct equation should be x^2 + y^2 = 8. So, your choice A is not correct.
The correct answer is C. x^2 - y^2 = 8.