# Thread: Help needed with martices and conics

1. ## Help needed with martices and conics

I'm not sure this is where I should put this, but I didn't see anywhere else:

1)

Using the determinant of a matrix representing a conic,
[a b d]
[b c e]
[d e f]
where the equation of the conic is
ax^2=2bxy+cy^2+2dx+2ey+f=0
and the determinant of the top left 2-by-2 matrix, how would you classify the conic?

2) (Refering to the d and e of the previous question)

Show that d^2+e^2 stays the same under a rotation of axes, in other words, the matrix is multiplied by
[cos(z) -sin(z) 0]
[sin(z) cos(z) 0]
[0 0 1]

Note: It's a 3x3 matrix

$b^2-ac \begin{cases} <0: \mbox{ellipse}\cr =0: \mbox{parabola} \cr >0: \mbox{hyperbola} \end{cases}$
2) Let $d', \ e'$ be the respective coefficients after the rotation. Calculate to get $d'=dcosz+esinz, \ d'=ecosz-dsinz.$ So $d'^2+e'^2=d^2+e^2$.