Morning, everyone,

I would like to know how a second-degree diophantine equation with two unknows like this, ax^2 + bxy + cy^2 + dx + ey + f = 0, where a,b,c,d,e and f are integers

can be transformed using linear transformation into this, mu^2 + nv^2 + g = 0, where m,n and g are integers

For example, 9x^2 + 6xy - 13y^2 - 6x - 16y + 20 = 0

There is : 2u^2 - 7v^2 + 45 = 0, where u = 3x + y - 1 and v = 2y + 1

Someone told me that mu^2 + nv^2 + g = 0 could be found using the linear transformation method. How do we get to mu^2 + nv^2 + g = 0 using this ?

Or is there another simpler method?

this term contain term xy which indicates that this equation is transformed into some angle with the axes. First we have to remove the xy term by rotating the curve by that angle through which it has been transformed.

So someone just helped me and indeed there are some very quick ways,