Question

The curve, for which the area of the triangle formed by X-axis, the tangent line at any point \(P\) and line \(OP\) is equal to \(a^2\), is given by

• A. \(y = x - Cx^2\)
• B. \(x = Cy \pm \frac{a^2}{y}\)
• C. \(y = Cx \pm \frac{a^2}{x}\)
• D. None of these

Hint:

Use intercept form of tangent to relate geometry with differential equations.

Answer 1

The Correct Option is B

Concept: Use tangent form and area of triangle.

Step 1: Tangent equation.
\[ y - y_1 = m(x - x_1) \] Intercept form gives triangle area.

Step 2: Area condition.
\[ \frac{1}{2} \times (\text{x-intercept}) \times (\text{y-intercept}) = a^2 \]

Step 3: Solve differential equation.
Leads to: \[ x = Cy \pm \frac{a^2}{y} \]