Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form.

Question

Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form. ​  ​

Quizplus · Accepted Answer

We can eliminate the xy-term by rotating the axes $45^\circ$ counterclockwise. This can be done by substituting $x' = x\cos(45^\circ) - y\sin(45^\circ)$ and $y' = x\sin(45^\circ) + y\cos(45^\circ)$, which simplifies to $x' = \frac{1}{\sqrt{2}}(x - y)$ and $y' = \frac{1}{\sqrt{2}}(x + y)$.

Substituting these expressions into the given equation, we get:

$11(x-\frac{1}{\sqrt{2}}y)^2 + 15(\frac{1}{\sqrt{2}}(x+y))^2 - 10(x-\frac{1}{\sqrt{2}}y)(\frac{1}{\sqrt{2}}(x+y)) - 66x - 66y + 198 = 0$

Simplifying, we get:

$11x^2+15y^2-132x+132y+88=0$

Dividing by $11$ and completing the square, we get:

$\frac{(x-6)^2}{3^2} + \frac{(y+4)^2}{2^2} = 1$

Therefore, the standard form of the equation is:

$\boxed{	extbf{(C) }11eacce6\_20b4\_deb7\_9f49\_6b1b26b3f8fb\_TB7896\_11}$

Rotate the Axes to Eliminate the Xy-Term in the Equation