To analyze the curve defined by the equation \(x^2 y^3 = C\), we need to find the tangent line at a point on the curve and determine how the segment of that tangent line between the x-axis and y-axis is divided by the point of tangency. Let's break this down step by step.
Understanding the Curve
The equation \(x^2 y^3 = C\) describes a relationship between \(x\) and \(y\) where \(C\) is a constant. This is a type of implicit function, and we can derive the slope of the tangent line by using implicit differentiation.
Finding the Derivative
To find the slope of the tangent line at a point \((x_0, y_0)\) on the curve, we differentiate both sides of the equation with respect to \(x\):
- Differentiate \(x^2\) to get \(2xy^3\).
- Differentiate \(y^3\) using the chain rule: \(3y^2 \frac{dy}{dx}\).
Setting the derivatives equal gives us:
2xy^3 + 3y^2 \frac{dy}{dx} x^2 = 0
Solving for \(\frac{dy}{dx}\) yields:
\(\frac{dy}{dx} = -\frac{2xy^3}{3y^2x^2}\)
Equation of the Tangent Line
At the point \((x_0, y_0)\), the slope is:
\(m = -\frac{2x_0y_0^3}{3y_0^2x_0^2} = -\frac{2y_0}{3x_0}\)
The equation of the tangent line can be expressed in point-slope form:
\(y - y_0 = m(x - x_0)\)
Finding Intercepts
To find where this tangent line intersects the axes, we can calculate the x-intercept and y-intercept:
X-Intercept
Set \(y = 0\) in the tangent line equation:
0 - y_0 = -\frac{2y_0}{3x_0}(x - x_0)
Solving for \(x\) gives us:
x = x_0 + \frac{3x_0}{2} = \frac{5x_0}{2}
Y-Intercept
Set \(x = 0\) in the tangent line equation:
y - y_0 = -\frac{2y_0}{3x_0}(0 - x_0)
Solving for \(y\) gives us:
y = y_0 + \frac{2y_0}{3} = \frac{5y_0}{3}
Ratio of Division
Now, we have the x-intercept \(\frac{5x_0}{2}\) and the y-intercept \(\frac{5y_0}{3}\). The point of tangency \((x_0, y_0)\) divides the segment between these intercepts.
To find the ratio in which the point of tangency divides the segment, we can use the section formula. The coordinates of the intercepts are:
- X-Intercept: \((\frac{5x_0}{2}, 0)\)
- Y-Intercept: \((0, \frac{5y_0}{3})\)
Using the section formula, we can find the ratio:
Let the point of tangency divide the segment in the ratio \(k:1\), where \(k\) is the ratio we want to find. The coordinates of the point dividing the segment are given by:
\(x = \frac{5x_0}{2} \cdot \frac{1}{k + 1} + 0 \cdot \frac{k}{k + 1}\)
\(y = 0 \cdot \frac{1}{k + 1} + \frac{5y_0}{3} \cdot \frac{k}{k + 1}\)
Setting these equal to \(x_0\) and \(y_0\) respectively, we can solve for \(k\). After solving, we find that the point of tangency divides the segment in the ratio of \(3:2\) internally.
Final Answer
Thus, the correct answer is that the portion of the tangent between the axes is divided by the point of tangency in the ratio of 3:2 internally.
