- Direction of asymptotes

- Calculating the directions of the asymptotes

- Asymptote of a conic section

- Calculating the asymptotes

- Theorem

- Quadratic equation of the asymptotes of an ellipse or hyperbola.

- Equation of a conic section with given asymptotes

F(x,y,z) = 3 x^{2}- y^{2}+ 2 xy + 4 x - 2 y + 7 = 0 Asymptote has slope m <=> (1,m,0) is on the conic section <=> 3 + 2 m - m^{2}= 0 <=> m = -1 or m = 3

- A parabola has two equal ideal points. It is THE asymptotic direction of the parabola.
- If a' = 0, the direction (0,1,0) is an asymptotic direction.
- The asymptotic directions do not depend on a".

- Calculate the ideal points
- Calculate the tangent lines

F(x,y,z) = 3 xThe ideal points are (1,-1,0) and (1,3,0).^{2}- y^{2}+ 2 xy + 4 x - 2 y + 7 = 0

The tangent line in (1,-1,0) is 3 x + y + 3 = 0

The tangent line in (1, 3 ,0) is 3 x - 3 y - 1 = 0

- The ideal line is the asymptote of a non-degenerated parabola.
- A degenerated parabola has a double point as ideal point. thus, each line through that point is an asymptote.
- The asymptotes of a degenerated ellipse or hyperbola are coinciding with the components.

FSo, we see that the asymtote does not depend on a" . This property can be useful to calculate the asymptotes._{x}' (x,y,z) + m F_{y}' (x,y,z) = 0 <=> ( a x + b" y + b' z ) + m ( b" x + a' y + b z ) = 0

Example:

x^{2}- xy - 2 x - 5 = 0 has the same asymptotes as x^{2}- xy - 2 x = 0 <=> x (x - y - 2) = 0 The asymptotes are x = 0 and x - y - 2 = 0

If two ellipsis or two hyperbolas have the same asymptotes, then their equations can be written such that only the a" differs.

x^{2}- x y - 2 y^{2}+ 3 x + 3 y + 7 = 0 It has the same asymptotes as x^{2}- x y - 2 y^{2}+ 3 x + 3 y + k = 0 Now, choose k such that the conic section is degenerated. The condition is DELTA = 0 <=> | 2, -1, 3 | | -1, -4, 3 | = 0 | 3, 3, 2 k | <=> -18 k = 0 <=> k = 0 Therefore, the quadratic equation of the asymptotes is x^{2}- x y - 2 y^{2}+ 3 x + 3 y = 0

Say uare the asymptotes of a conic section._{1}x + v_{1}y + w_{1}= 0 u_{2}x + v_{2}y + w_{2}= 0

The degenerated conic section with these asymptotes is

(uAll conic sections with these asymptotes have equation_{1}x + v_{1}y + w_{1})(u_{2}x + v_{2}y + w_{2}) = 0

(u_{1}x + v_{1}y + w_{1})(u_{2}x + v_{2}y + w_{2}) + h = 0

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