WBJEE · Maths · Indefinite Integration

A family of curves is such that the length intercepted on the \(y\) -axis between the origin and the tangent at a point is three times the ordinate of the point of contact. The family of curves is

A \(x=C\), \(C\) is a constant
B \(x y^{2}=C, C\) is a constant
C \(x^{2} y=C, C\) is a constant
D \(x^{2} y^{2}=C, C\) is a constant

Verified Solution

Answer & Solution

Correct Answer

(C) \(x^{2} y=C, C\) is a constant

Step-by-step Solution

Detailed explanation

Let the general equation of tangent. Which passes through the point \((x, y)\) is \((Y-y)=\frac{d y}{d x}(x-x)\) \(\Rightarrow \quad Y-y=X \frac{d y}{d x}-x \frac{d y}{d x}\) for length of \(Y\) -intercept, put \(X=0\) in Eq. \((1),\) we get \(Y-y=-x \frac{d y}{d x}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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