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= 3 - 3 =0
4 4
The given equation represents a degenerate conic which is a pair of lines. The given

equation is

2x2 + x(5 - y) + (-2y + 2) = 0

or x = y - 5 ± ( y - 5)2 - 8(-2 y + 2)
4

= y - 5 ± y2 -10 y + 25 +16 y -16
4

= y - 5 ± ( y + 3)
4

= 2y -2, -2
4

Equations of the lines are 2x - y + 1 = 0 and x + 2 = 0.

Tangent (1)
Find an equation of the tangent to the conic
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
at the point (x1, y1)
Differentiating (1) w.r.t. x, we have

2ax + 2hy + 2hx dy + 2by dy + 2g + 2 f dy =0
dx dx dx

or dy = - ax + hy + g
dx hx + by + f

or dy  = - ax1 + hy1 + g
dx  hx1 + by1 + f
( x1 , y1 )

Equation of the tangent at (x1, y1) is

y - y1 =- ax1 + hy1 + g ( x1 , y1)
hx1 + by1 + f

or (x - x1)(ax1 + hy1 + g) + ( y - y1)(hx1 + by1 + f ) =0
or axx1 + hxy1 + gx + +hx1y + by1y + fy

version: 1.1

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