CUET · MATHS · PYQ PAPER 2023

The two curves \(x^3-3 x y^2+15=0\) and \(3 x^2 y-y^3+17=0\) :

A cut at right angles
B touch each other
C cut at an angle \(\pi / 4\)
D cut at an angle \(\pi / 3\)

Verified Solution

Answer & Solution

Correct Answer

(A) cut at right angles

Step-by-step Solution

Detailed explanation

\(m_1 = \frac{d}{dx}(x^3 - 3xy^2 + 15=0) = \frac{3x^2 - 3y^2}{6xy} = \frac{x^2 - y^2}{2xy}\) \(m_2 = \frac{d}{dx}(3x^2y - y^3 + 17=0) = \frac{-6xy}{3x^2 - 3y^2} = \frac{-2xy}{x^2 - y^2}\) \(m_1 m_2 = \left(\frac{x^2 - y^2}{2xy}\right) \left(\frac{-2xy}{x^2 - y^2}\right) = -1\)…

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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