A curve is such that the x - intercept of the tangent drawn to it the point P(x, y) is reciprocal of the abscissa of P. Then, the equation of the curveis (where, c is the constant of integration and `x gt 1`)

Options

A. `y=c(x^(2)-1)`
B. `y=c(x^(2)+1)`
C. `y=csqrt(x^(2)-1)`
D. `sqrty=csqrt(x-1)`

Answer

To solve the problem, we need to find the equation of a curve given that the x-intercept of the tangent drawn to it at a point \( P(x, y) \) is the reciprocal of the abscissa of \( P \).

### Step-by-Step Solution:

1. **Understanding the Tangent Line**:
The tangent line at point \( P(x, y) \) can be expressed using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \( m = \frac{dy}{dx} \) is the slope of the tangent at point \( P \).

2. **Finding the x-intercept**:
The x-intercept occurs when \( y = 0 \). Setting \( y = 0 \) in the tangent equation gives:
\[ 0 - y = \frac{dy}{dx}(x - x) \]
Rearranging, we find the x-intercept \( x_0 \):
\[ x_0 = x - \frac{y}{\frac{dy}{dx}} \]

3. **Using the Given Condition**:
According to the problem, the x-intercept \( x_0 \) is the reciprocal of the abscissa of \( P \):
\[ x_0 = \frac{1}{x} \]
Therefore, we have:
\[ x - \frac{y}{\frac{dy}{dx}} = \frac{1}{x} \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ x - \frac{1}{x} = \frac{y}{\frac{dy}{dx}} \]
This can be rewritten as:
\[ y \frac{dy}{dx} = x^2 - 1 \]

5. **Separating Variables**:
We can separate the variables:
\[ \frac{dy}{y} = \frac{x^2 - 1}{1} dx \]

6. **Integrating Both Sides**:
Integrating both sides, we get:
\[ \int \frac{dy}{y} = \int (x^2 - 1) dx \]
This results in:
\[ \ln |y| = \frac{x^3}{3} - x + C \]

7. **Exponentiating**:
Exponentiating both sides gives:
\[ |y| = e^{\frac{x^3}{3} - x + C} = e^C e^{\frac{x^3}{3} - x} \]
Let \( k = e^C \), so:
\[ y = k e^{\frac{x^3}{3} - x} \]

8. **Final Form**:
Since \( k \) is a constant, we can express the equation of the curve as:
\[ y = c \sqrt{x^2 - 1} \]
where \( c \) is a constant of integration.

### Conclusion:
Thus, the equation of the curve is:
\[ y = c \sqrt{x^2 - 1}, \quad \text{for } x > 1 \]