Question

does euqation of pair of lines have a xy term ? a second degree equation has xy term what about a pair of lines or pair of circles?

Answer 1

A pair of lines represented by :
                    L1 : ax + by + c = 0  ,  slope = m1 = -a/b
                and  L2 = d x + e y + f = 0        slope  m2 = - d/e

 L1L2:     (a x + b y + c) (d x + e y + f) = 0

L1L2:   a d x² + b e y² + (b d + a e) x y + (a f + c d) x + (b f + c e) y + c f  = 0

So a pair of straight lines has a XY term.

    =  (ad/be) x² + y² + (d/e + a/b) x y + (af + cd)/be  x + (f/e + c/b) y + cf/ be = 0
   =  m1 m2 x² + y² - (m1 + m2) x y + ( af + cd)/be x +  (f/e + c/b) y  + cf/be = 0
in this form, the coefficient of x y is the sum of slopes of the lines.
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a circles:

   generally  for a circle, the equation is :    (x - a)²  +  (y - b)² = r²

  This does not contain a term in XY.