Rotation and translation in Analytical geometry

Question

The equation of a curve is 11x² + 2√3 xy + 9y²= 12(x√3+ y + 1) in a rectangular Cartesian system. We first rotate the axes through 30 ^° about the origin, what do we get?

 

(A) Let (X,Y) be the new coordinates of the curve when rotated through 30^° about the origin. 

So, x= 1/2(√3X - Y) and y = 1/2( X + Y√3)

By substituting the values of x and y in the equation of the given curve we get,

11/4(X√3 - Y)² + (2√3/4)(X√3 - Y)(X + Y√3) + 9/4(X + Y√3)² = 12{ [(√3/2)(X√3 - Y)] + [(1/2)(X + Y√3)] + 1}

Solving the above equation, I got, 

12 X² + 8Y² = 12 (2X+1) 

But the key as given in the Text book is 

12X (X -1) + 8Y² = 3. 

Can you please tell me which is the correct answer among the two? And thank you for your quick response.

Answer 1

The equation is of a rotated ellipse.

Now is this ellipse rotated another 30 degrees?

Or is it the results of a rotation of 30 degrees?

Proof your statement, edit, and resubmit.