Orthogonal transformations done right?

Question

Q. Transform the equation $2x^2-xy+y^2+2x-3y+5=0$ to new axes given of $x$ and $y$ given by the straight lines $4x+3y+1=0$ and $3x-4y+2=0$ respectively.

My Attempt:

We first find the new origin by simultaneously solving the given equations of the two straight lines. The intersection pt of those two lines (aka the new origin) is $(-2/5,1/5)$.

Applying translation as follows :

$$2(x-\frac 25)^2-(x-\frac 25)(y+\frac 15)+(y+\frac 15)^2+2(x-\frac 25)-3(y+\frac 15)+5=0$$

Now, after translation, the new equation looks like $50x^2-25xy+25y^2+5x-55y+101=0$.

For rotation, if the angle to be rotated is $\theta$, we can easily note that $\tan\theta$ is the slope of the straight line given as new $x$-axis.

So, $\tan\theta=\frac{-4}3$. Then we get $\sin\theta=4/5$ and $\cos\theta=-3/5$

Now, we apply rotation as follows :

$$50(-\frac 35 x-\frac 45 y)^2-25(-\frac 35 x-\frac 45 y)(\frac 45 x-\frac 35 y)+25(\frac 45 x-\frac 35 y)^2+5(-\frac 35 x-\frac 45 y)-55(\frac 45 x-\frac 35 y)+101=0$$

After rotation, the final transformed equation seems to be $30x^2+20y^2+31xy-31x+38y+101=0$.

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The problem is that my book has a different answer. It says the answer is $46x^2+29y^2+31xy+47x-29y+101=0$.

I might have overlooked some calculation mistakes. Can someone take a look at this? Thanks.

Answer 1

As discussed in the comments, there was an incorrect substitution (now edited) of $-\frac35x-\frac45x$ instead of $-\frac35x-\frac45y$ for $x$, but the transformed equation corresponded to neither version and seems to have been the result of calculation errors. When the calculation is performed correctly, the result is in line with the one given in the book, except for the signs of the new $x$ and $y$ coordinates, which are of course arbitrary since only the new axes and not their directions were given.

Some room for improvement: The whole thing would be somewhat more structured (and thus less error-prone) if you wrote it in the form

$$ \vec r^\top A\vec r+\vec b^\top\vec r+c=0 $$

and then applied the translation as $\vec r\to\vec r-\vec r_0$ and the rotation as $\vec r\to R\vec r$. Also, using \left and \right before opening and closing parentheses (or any opening and closing delimiters) makes them adapt to the size of the content.