Quantitative Aptitude Ques 274

Question: If $ x=a\sin \theta -b\cos \theta , $ $ y=a\cos \theta +b\sin \theta , $ then which of the following is true?

Options:

A) $ x^{2}+y^{2}=a^{2}+b^{2} $

B) $ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $

C) $ x^{2}+y^{2}=a^{2}-b^{2} $

D) $ \frac{x^{2}}{y^{2}}+\frac{a^{2}}{b^{2}}=1 $

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Answer:

Correct Answer: A

Solution:

  • We have, $ x=a\sin \theta -b\cos \theta $

$ \therefore $ $ x^{2}=a^{2}{{\sin }^{2}}\theta +b^{2}\cos \theta -2ab\sin \theta \cdot \cos \theta $ Similarly, $ y^{2}=a^{2}{{\cos }^{2}}\theta +b^{2}{{\sin }^{2}}\theta +2ab\sin \theta \cdot \cos \theta $

$ \therefore $ $ x^{2}+y^{2}=a^{2}+b^{2} $