Question

The differential equation of all straight lines passing through the point \( (1, -1) \) is

• A. \( y = (x - 1)\frac{dy}{dx} - 1 \)
• B. \( x = (x - 1)\frac{dy}{dx} + 1 \)
• C. \( y = (x - 1)\frac{dy}{dx} \)
• D. \( y = 2(x - 1)\frac{dy}{dx} \)

Hint:

Differentiate the family of curves and eliminate the arbitrary constant to get the differential equation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to find the differential equation representing a family of curves.
First, we write the general equation of the family of lines passing through a specific point. This equation will contain an arbitrary constant (the slope).
Then, we eliminate the arbitrary constant by differentiating the equation.
Step 2: Key Formula or Approach:
The equation of a straight line passing through a point \( (x_1, y_1) \) with slope \( m \) is \( y - y_1 = m(x - x_1) \).
Differentiate with respect to \( x \) to find an expression for \( m \), and substitute it back into the original equation.
Step 3: Detailed Explanation:
The given point is \( (1, -1) \). Let the slope of the line be \( m \).
The equation of the family of lines is:
\[ y - (-1) = m(x - 1) \] \[ y + 1 = m(x - 1) \quad \dots \text{ (Equation 1)} \] To form the differential equation, we need to eliminate the parameter \( m \).
Differentiate Equation 1 with respect to \( x \):
\[ \frac{d}{dx}(y + 1) = \frac{d}{dx}[m(x - 1)] \] \[ \frac{dy}{dx} = m \cdot (1) + 0 \] \[ m = \frac{dy}{dx} \] Now, substitute this value of \( m \) back into Equation 1:
\[ y + 1 = \left( \frac{dy}{dx} \right) (x - 1) \] Rearranging to match the options:
\[ y = (x - 1)\frac{dy}{dx} - 1 \] Step 4: Final Answer:
The correct differential equation is \( y = (x - 1)\frac{dy}{dx} - 1 \).