The curve passing through `P(2, (7)/(2))` and having place `1-(1)/(x^(2))` at P (x,y) also passes through

Options

A. `(-2, (2)/(3))`
B. `(-2, -(3)/(2))`
C. `(-2, 1)`
D. `(-2, 6)`

Answer

To solve the problem, we need to find the equation of the curve given its derivative and then determine which points satisfy this equation. Let's go through the solution step by step.

### Step 1: Write down the given derivative
The derivative of the curve is given as:
\[ \frac{dy}{dx} = 1 - \frac{1}{x^2} \]

### Step 2: Integrate the derivative to find the equation of the curve
To find the equation of the curve, we need to integrate the derivative:
\[ dy = \left(1 - \frac{1}{x^2}\right) dx \]
Integrating both sides:
\[ y = \int \left(1 - \frac{1}{x^2}\right) dx \]
This can be split into two integrals:
\[ y = \int 1 \, dx - \int \frac{1}{x^2} \, dx \]
Calculating the integrals:
\[ y = x + \frac{1}{x} + C \]
where \(C\) is the constant of integration.

### Step 3: Use the point \(P(2, \frac{7}{2})\) to find \(C\)
We know that the curve passes through the point \(P(2, \frac{7}{2})\). We can substitute \(x = 2\) and \(y = \frac{7}{2}\) into the equation:
\[ \frac{7}{2} = 2 + \frac{1}{2} + C \]
Simplifying the right side:
\[ \frac{7}{2} = 2 + 0.5 + C \]
\[ \frac{7}{2} = \frac{4}{2} + \frac{1}{2} + C \]
\[ \frac{7}{2} = \frac{5}{2} + C \]
Now, solving for \(C\):
\[ C = \frac{7}{2} - \frac{5}{2} = 1 \]

### Step 4: Write the final equation of the curve
Now that we have \(C\), we can write the equation of the curve:
\[ y = x + \frac{1}{x} + 1 \]

### Step 5: Check which points satisfy the equation
We need to check which of the given points satisfies the equation. Let's check the point \((-2, -\frac{3}{2})\):
Substituting \(x = -2\) into the equation:
\[ y = -2 + \frac{1}{-2} + 1 \]
Calculating:
\[ y = -2 - \frac{1}{2} + 1 = -2 - 0.5 + 1 = -1.5 = -\frac{3}{2} \]
This shows that the point \((-2, -\frac{3}{2})\) satisfies the equation.

### Conclusion
The curve passes through the point \((-2, -\frac{3}{2})\).