Rotation Of Axes

Step 1: Concept
For a rotation

%5Ctheta

, the transformations are

x%20%3D%20X%5Ccos%5Ctheta%20-%20Y%5Csin%5Ctheta

and

y%20%3D%20X%5Csin%5Ctheta%20%2B%20Y%5Ccos%5Ctheta

.
Step 2: Analysis
Given

%5Ctan%5Ctheta%20%3D%202

, we have

%5Ccos%5Ctheta%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D%7D

and

%5Csin%5Ctheta%20%3D%20%5Cfrac%7B2%7D%7B%5Csqrt%7B5%7D%7D

. Thus,

x%20%3D%20%5Cfrac%7BX-2Y%7D%7B%5Csqrt%7B5%7D%7D

and

y%20%3D%20%5Cfrac%7B2X%2BY%7D%7B%5Csqrt%7B5%7D%7D

.
Step 3: Evaluation
Substituting these into

4xy%20-%203x%5E%7B2%7D%20%3D%20a%5E%7B2%7D

4(%5Cfrac%7BX-2Y%7D%7B%5Csqrt%7B5%7D%7D)(%5Cfrac%7B2X%2BY%7D%7B%5Csqrt%7B5%7D%7D)%20-%203(%5Cfrac%7BX-2Y%7D%7B%5Csqrt%7B5%7D%7D)%5E%7B2%7D%20%3D%20a%5E%7B2%7D

%5Cfrac%7B4(2X%5E%7B2%7D%20-%203XY%20-%202Y%5E%7B2%7D)%7D%7B5%7D%20-%20%5Cfrac%7B3(X%5E%7B2%7D%20-%204XY%20%2B%204Y%5E%7B2%7D)%7D%7B5%7D%20%3D%20a%5E%7B2%7D

.
Step 4: Conclusion
Simplifying gives

5X%5E%7B2%7D%20-%2020Y%5E%7B2%7D%20%3D%205a%5E%7B2%7D

, which reduces to

X%5E%7B2%7D%20-%204Y%5E%7B2%7D%20%3D%20a%5E%7B2%7D

.
Final Answer: (a)

Chapter Questions
1 MCQs