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 (4, 1) is a point on the curve?
x2 − y2 = 15
x2 + y2 = 15
x + y = 15
xy = 15
To find the equation of the curve, we need to use the given information, which is the slope of the tangent line at any point (x, y) on the curve is x/y.
The slope of the tangent line at a point (x, y) represents the derivative of the function at that point. In other words, we can write the derivative as dy/dx = x/y.
To find the equation of the curve, we can integrate this derivative equation. Let's separate the variables and integrate:
dy/dx = x/y
y dy = x dx
Integrating both sides:
∫y dy = ∫x dx
1/2 y^2 = 1/2 x^2 + C
Now, we can use the given point (4, 1) to find the value of C:
1/2 (1)^2 = 1/2 (4)^2 + C
1/2 = 8 + C
C = -7.5
Thus, the equation of the curve can be written as:
1/2 y^2 = 1/2 x^2 - 7.5