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: Concept General equation of a line passing through \((x_1, y_1)\) is \(y - y_1 = m(x - x_1)\).

Step 2: Meaning Here \((x_1, y_1) = (1, -1)\). Equation: \(y - (-1) = m(x - 1) \rightarrow y + 1 = m(x - 1)\).

Step 3: Analysis To find the differential equation, differentiate with respect to \(x\): \(\frac{dy}{dx} = m\). Substitute \(m\) back into the line equation: \(y + 1 = \frac{dy}{dx}(x - 1)\).

Step 4: Conclusion Rearranging gives \(y = (x - 1)\frac{dy}{dx} - 1\). Final Answer: (A)