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

Question

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

Answer 1

Maybe I should become a politician.

Never actually answer the question, rather give an answer to some other question.
Thanks for the catch Steve, that is two for one evening, I should get more sleep.

Answer 2

slope = x/y = 3/1 = 3

so the equation of the tangent at the point (3,1) is
y-1 = 3(x-3)
y = 3x -8 or 3x - y - 8 = 0

Answer 3

y' = x/y

you can recognize that as an hyperbola

x^2-y^2 = a^2
since (3,1) is on the curve,
9-1 = a^2

x^2-y^2 = 8

Answer 4

Well, it seems like this curve is a bit of a math joker! The equation should make us giggle a bit.

Let's assume that the equation of the curve is y = f(x), then we know that the slope of the tangent line at any point (x, y) on the curve is x divided by y. In other words, the derivative dy/dx is equal to x / y.

Now, let's integrate both sides with respect to x and see what kind of fun equation we end up with:

∫ (1/y) dy = ∫ x dx

ln|y| = (1/2)x^2 + C

Where C is the constant of integration. But wait, the equation needs to pass through the point (3, 1). Let's use this information to find the value of C.

ln|1| = (1/2)(3)^2 + C

0 = 9/2 + C

C = -9/2

So, the equation of this hilarious curve would be:

ln|y| = (1/2)x^2 - 9/2

Answer 5

To find the equation of the curve, we need to determine its slope at any given point (x, y) on the curve and then integrate it to find the equation.

Given that the slope of the tangent line at any point (x, y) on the curve is x/y, we can express this relationship as:

dy/dx = x/y

To separate the variables, we can multiply both sides of the equation by y:

y * dy/dx = x

Now, we can integrate both sides with respect to x to eliminate the derivative:

∫ y * dy = ∫ x dx

Integrating both sides gives us:

(1/2) * y^2 = (1/2) * x^2 + C

Where C is the constant of integration.

To find the value of C, we can use the given point (3, 1) as it lies on the curve. Substituting x = 3 and y = 1 into the equation, we get:

(1/2) * 1^2 = (1/2) * 3^2 + C
1/2 = 9/2 + C

Simplifying the equation:

1/2 - 9/2 = C
-4 = C

Therefore, the equation of the curve is:

(1/2) * y^2 = (1/2) * x^2 - 4