Question

If \[x=a\cos \theta \] and \[y=b\sin \theta \], then \[{{b}^{2}}{{x}^{2}}+{{a}^{2}}{{y}^{2}}\] is equal to
A) $ab$
B) \[{{a}^{2}}{{b}^{2}}\]
C) \[{{a}^{4}}{{b}^{4}}\]
D) None of these

Answer 1

Hint: In the given question, we have been asked to find the value of the given trigonometric expression. In order to find the value of a given expression, first we need to simplify the given expression so that we can apply the trigonometric identity. After rewritten apply the trigonometric identity that is \[{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1\]. Later substituting the values, solve the given expression and then find the value of a function by using the calculator. In this way we will get the required answer.

Complete step-by-step solution:
We have given that,
\[x=a\cos \theta \] and \[y=b\sin \theta \]
Now,
\[\Rightarrow x=a\cos \theta \]
Therefore,
\[\Rightarrow \cos \theta =\dfrac{x}{a}\]
Then,
We have,
\[\Rightarrow y=b\sin \theta \]
Therefore,
\[\Rightarrow \sin \theta =\dfrac{y}{b}\]
Using the trigonometric identity i.e.
\[\Rightarrow {{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1\]
Therefore,
Substituting the values, we obtained
\[\Rightarrow \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\]
Solving the above expression, by taking the LCM, we get
\[\Rightarrow \dfrac{{{x}^{2}}{{b}^{2}}+{{y}^{2}}{{a}^{2}}}{{{a}^{2}}{{b}^{2}}}=1\]
Multiplying both the sides of the equation by \[{{a}^{2}}{{b}^{2}}\], we get
\[\Rightarrow {{x}^{2}}{{b}^{2}}+{{y}^{2}}{{a}^{2}}={{a}^{2}}{{b}^{2}}\]
Therefore, the value of \[{{b}^{2}}{{x}^{2}}+{{a}^{2}}{{y}^{2}}\]is equal to \[{{a}^{2}}{{b}^{2}}\]. It is the required answer.

Hence the option (B) is the correct answer.

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometric and the trigonometric ratios as well. It will make questions easier to solve. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.