General Equation of a Conic

Hello rebirthflame Quote:

Originally Posted by

**rebirthflame** im not really sure if this is to do with quadratic forms or not but tbh i have no idea of where to start with this question, i think it has something to do with diagonilising a matrix somewhere along the line but apart from that im clueless

by using suitable transformations express

$\displaystyle x^2+x-8+5xy-6y+2y^2 = 0$

in the form

$\displaystyle AX^2+BY^2=1$

What you are being asked to do here is to transform a general equation of the second degree into a standard form for a conic. The usual method, starting with

$\displaystyle ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$

is to let

$\displaystyle x = X\cos\theta - Y\sin\theta$

$\displaystyle y = X\sin\theta + Y\cos\theta$

which corresponds to a rotation of the axes through an angle $\displaystyle \theta$, in order to eliminate the term in $\displaystyle xy$. When you do this, it leads to the value of $\displaystyle \theta$ given by

$\displaystyle \tan 2\theta = \frac{2h}{a-b}$

You then complete the squares for $\displaystyle X$ and $\displaystyle Y$ to get the equation in the required form.

In the example you've been given:

$\displaystyle \tan 2\theta = -5$

But this doesn't seem to give straightforward (i.e. easy) values for $\displaystyle \sin\theta$ and $\displaystyle \cos\theta$ ($\displaystyle \sin\theta = \sqrt{\frac{1 + \sqrt{26}}{2\sqrt{26}}}$, for instance).

Are you sure you have the right numbers here?

Grandad