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 (2, 1) is a point on the curve?
To find the equation of a curve, we need to integrate the given slope function. Let's do that step by step.
Given: The slope of the tangent line to the curve at any point (x, y) is x/y.
Step 1: Find the equation of the tangent line at some point (x, y).
The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. Given that the slope is x/y, we have:
y = (x/y)x + b
Simplifying, we get:
y = (x^2)/y + b
Step 2: Solve for b.
We know that the point (2, 1) lies on the curve, which means it also lies on the tangent line. Substituting these values into the equation, we have:
1 = (2^2)/1 + b
1 = 4 + b
b = 1 - 4
b = -3
Step 3: Substitute the value of b to get the final equation.
Substituting the value of b = -3 into the equation, we have:
y = (x^2)/y - 3
So, the equation of the curve is y = (x^2)/y - 3.
To find the equation of the curve, we need to integrate the given slope. Let's proceed step by step.
Step 1: Find the derivative of the curve equation.
The slope of a curve at any point (x, y) is given as the derivative dy/dx. In this case, dy/dx = x/y.
Step 2: Rewrite the derivative equation.
Multiplying both sides of the derivative equation by y, we get dy = x * dx.
Step 3: Integrate both sides of the equation.
Integrating both sides, we have ∫dy = ∫x * dx.
Step 4: Evaluate the integrals.
On the left side, the integral of dy is y. On the right side, the integral of x with respect to x is x^2/2.
So, the equation becomes y = x^2/2 + C, where C is the constant of integration.
Step 5: Determine the value of C using the given point.
To find the value of C, we can substitute the given point (2, 1) into the equation. So, when x = 2 and y = 1, we have:
1 = (2^2)/2 + C
1 = 4/2 + C
1 = 2 + C
Subtracting 2 from both sides, we get C = -1.
Step 6: Write the final equation.
Now that we know the value of C, we can substitute it back into the equation to get the final equation:
y = x^2/2 - 1
Therefore, the equation of the curve is y = x^2/2 - 1, with the given point (2, 1) on the curve.
dy/dx = x/y
y dy = x dx
y^2 = x^2 + c
using the point (2,1)
1 = 4+c
c = -3
x^2-y^2 = 3