A tangent to a curve at P(x, y) intersects x-axis and y-axis at A and B respectively. Let the point of contact divides AB in the ratio `y^2 : x^2`.
If a member of this family passes through (3, 4) then the equation of curve and area of the curve is

Options

A. `((25)/(4) + (2pi)/(3))`
B. `2(pi + 4)`
C. `25pi`
D. `(16pi + (4)/(5))`

Answer

To solve the problem step-by-step, we will derive the equation of the curve and then calculate the area under the curve.

### Step 1: Write the equation of the tangent line
The equation of the tangent to the curve at the point \( P(x, y) \) can be expressed as:
\[ Y - y = \frac{dy}{dx}(X - x) \]
where \( \frac{dy}{dx} \) is the slope of the tangent at point \( P \).

### Step 2: Find the intersection points A and B
1. **Intersection with the x-axis (Point A)**: Set \( Y = 0 \):
\[ 0 - y = \frac{dy}{dx}(X - x) \implies X = x - \frac{y}{\frac{dy}{dx}} \]
Thus, the coordinates of point A are:
\[ A\left(x - \frac{y}{\frac{dy}{dx}}, 0\right) \]

2. **Intersection with the y-axis (Point B)**: Set \( X = 0 \):
\[ Y - y = \frac{dy}{dx}(0 - x) \implies Y = y - x \frac{dy}{dx} \]
Thus, the coordinates of point B are:
\[ B\left(0, y - x \frac{dy}{dx}\right) \]

### Step 3: Use the section formula
The point \( P(x, y) \) divides the line segment \( AB \) in the ratio \( y^2 : x^2 \). Using the section formula:
\[ P = \left(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\right) \]
where \( m_1 = y^2 \), \( m_2 = x^2 \), \( (x_1, y_1) = (0, y - x \frac{dy}{dx}) \), and \( (x_2, y_2) = \left(x - \frac{y}{\frac{dy}{dx}}, 0\right) \).

### Step 4: Set up the equations
From the x-coordinate:
\[ x = \frac{y^2 \cdot \left(x - \frac{y}{\frac{dy}{dx}}\right) + x^2 \cdot 0}{y^2 + x^2} \]
This simplifies to:
\[ x(y^2 + x^2) = y^2 \left(x - \frac{y}{\frac{dy}{dx}}\right) \]

From the y-coordinate:
\[ y = \frac{y^2 \cdot 0 + x^2 \cdot \left(y - x \frac{dy}{dx}\right)}{y^2 + x^2} \]
This simplifies to:
\[ y(y^2 + x^2) = x^2 \left(y - x \frac{dy}{dx}\right) \]

### Step 5: Derive the differential equation
From the x-coordinate equation, we can rearrange and simplify to obtain:
\[ xy^2 = -x^2 y \frac{dy}{dx} \]
Dividing both sides by \( xy \) (assuming \( x, y \neq 0 \)):
\[ \frac{dy}{dx} = -\frac{y^2}{x^2} \]

### Step 6: Solve the differential equation
This is a separable differential equation:
\[ \frac{dy}{y^2} = -\frac{dx}{x^2} \]
Integrating both sides:
\[ -\frac{1}{y} = \frac{1}{x} + C \]
Rearranging gives:
\[ \frac{1}{y} + \frac{1}{x} = C \]
Multiplying through by \( xy \):
\[ x + y = Cxy \]
This can be rearranged to:
\[ Cxy - x - y = 0 \]

### Step 7: Find the constant C using the point (3, 4)
Substituting \( (x, y) = (3, 4) \):
\[ C(3)(4) - 3 - 4 = 0 \implies 12C - 7 = 0 \implies C = \frac{7}{12} \]

### Step 8: Write the equation of the curve
The equation becomes:
\[ \frac{7}{12}xy - x - y = 0 \implies 7xy - 12x - 12y = 0 \]

### Step 9: Calculate the area of the curve
The area under the curve can be calculated using the formula for the area of a circle since the derived equation represents a circle:
\[ \text{Area} = \pi r^2 \]
From the derived equation, we can find the radius:
\[ r^2 = \left(\frac{12}{7}\right)^2 \implies r = \frac{12}{7} \]
Thus, the area is:
\[ \text{Area} = \pi \left(\frac{12}{7}\right)^2 = \frac{144\pi}{49} \]

### Final Answer
The equation of the curve is:
\[ 7xy - 12x - 12y = 0 \]
The area of the curve is:
\[ \frac{144\pi}{49} \]