Tangent is drawn at any point P of a curve which passes through `(1, 1)` cutting x-axis and y-axis at A and B respectively. If `AP: BP = 3:1`, then,

Options

A. (a)Differential equation of the curve is `3x(dy)/(dx) + y = 0`
B. (b)Differential equation of the curve is `3x(dy)/(dx) - y = 0`
C. (c)Curve is passing through `((1)/(8), 2)`
D. (d)Normal at (1, 1) is x+3y = 4

Answer

To solve the problem step by step, we will derive the equation of the curve based on the given conditions.

### Step 1: Understand the problem
We have a curve that passes through the point (1, 1). A tangent is drawn at a point P on the curve, which intersects the x-axis at point A and the y-axis at point B. The ratio of the segments AP and BP is given as 3:1.

### Step 2: Set up the tangent line
Let the coordinates of point P be (x, y). The equation of the tangent line at point P can be expressed using the point-slope form:

\[ y - y_1 = m(x - x_1) \]

where \(m\) is the slope of the tangent at point P, which is given by \(\frac{dy}{dx}\) at that point.

### Step 3: Find the intercepts A and B
The x-intercept (A) occurs when \(y = 0\):

\[ 0 - y = m(x - x) \implies x = \frac{y}{m} + x \]

The y-intercept (B) occurs when \(x = 0\):

\[ y - y = m(0 - x) \implies y = mx + y \]

### Step 4: Use the ratio of segments
Given that \(AP:BP = 3:1\), we can express this in terms of the intercepts. Let \(AP = 3k\) and \(BP = k\) for some constant \(k\). The total length of AB is:

\[ AB = AP + BP = 3k + k = 4k \]

### Step 5: Relate the intercepts to the ratio
From the intercepts, we can express the lengths in terms of the coordinates of A and B. If A is at \((\frac{y}{m}, 0)\) and B is at \((0, mx)\), we can write:

\[ AP = \frac{y}{m} - x \quad \text{and} \quad BP = mx - y \]

### Step 6: Set up the equation from the ratio
From the ratio \(AP:BP = 3:1\), we have:

\[ \frac{\frac{y}{m} - x}{mx - y} = 3 \]

Cross-multiplying gives:

\[ \frac{y}{m} - x = 3(mx - y) \]

### Step 7: Simplify the equation
Expanding and simplifying:

\[ \frac{y}{m} - x = 3mx - 3y \]
\[ \frac{y}{m} + 3y = 3mx + x \]
\[ \frac{y + 3my}{m} = (3m + 1)x \]

### Step 8: Rearranging to find the derivative
Rearranging gives us a differential equation. We can express it as:

\[ dy + 3m dy = (3m + 1)dx \]

### Step 9: Solve the differential equation
This leads to:

\[ \frac{dy}{dx} = -\frac{3x}{y} \]

### Step 10: Separate variables and integrate
Separating variables gives:

\[ y \, dy = -3x \, dx \]

Integrating both sides:

\[ \frac{y^2}{2} = -\frac{3x^2}{2} + C \]

### Step 11: Find the constant C using the point (1, 1)
Substituting \(x = 1\) and \(y = 1\):

\[ \frac{1^2}{2} = -\frac{3(1^2)}{2} + C \implies \frac{1}{2} = -\frac{3}{2} + C \implies C = 2 \]

### Step 12: Final equation of the curve
Thus, the equation becomes:

\[ y^2 + 3x^2 = 4 \]

### Conclusion
The equation of the curve is \(y^2 + 3x^2 = 4\).