Second-Order Curves Given by an Equation in General Form

Determine the type of the curves given by the second-order equations:

Determine the type of the curves given by the second-order equations:

Determine $k$ so that the equation $x^{2}+6xy+y^{2}+6x+2y+k=0$ represents a pair of intersecting lines.

Determine $k$ so that the equation $3x^{2}-2xy+3y^{2}-2x+2y+k=0$ represents a point.

Choose $\lambda$ so that the equation $x^{2}-2\lambda xy+4y^{2}+2x+\lambda y=0$ represents a pair of lines, and find them.

In the equation $2x^{2}+\lambda xy+2y^{2}-7x+\mu y+3=0$, choose the coefficients $\lambda$ and $\mu$ so that the equation represents a pair of parallel lines.

Find the equations of the asymptotes of the curve $2x^{2}-xy+3x+y-1=0$.

Show that the equation $(ax+by+c)(a_{1}x+b_{1}y+c_{1})+\lambda =0$, where $\lambda \neq 0$ and $ab_{1}-a_{1}b \neq 0$, represents a hyperbola with asymptotes $ax+by+c=0$, $a_{1}x+b_{1}y+c_{1}=0$.

Find the equation of the hyperbola having as asymptotes the lines $x-1=0$ and $y-1=0$ and passing through the point $(2,2)$.

Prove that for any values of the coefficients $\lambda$ and $\mu$, the curve $\lambda f_{1}+\mu f_{2}=0$ passes through the points of intersection of the curves $f_{1}=0$ and $f_{2}=0$. Here $f_{1}$ and $f_{2}$ are polynomials in $x$ and $y$.

Let $s_{kl}=0$ be the equation of the line $A_{kl}x+B_{kl}y+C_{kl}=0$ passing through the points $k$ and $l$. Prove that for any $\lambda$ and $\mu$, the equation $\lambda s_{12}s_{34}+\mu s_{13}s_{24}=0$ represents a second-order curve passing through the four given points.

Find the equation of the second-order curve passing through the points $(1,1)$, $(2,-1)$, $(1,-2)$, $(-1,1)$, $(3,0)$.

Find the second-order curve from five of its points: $(3,1)$, $(2,1)$, $(-7,1)$, $(-2,0)$, $(0,1)$.

Find the parabola passing through the four points: $(0,-1)$, $(0,3)$, $(-1,0)$, $(3,0)$.

Find the second-order curve passing through the point $(1,1)$ and the points of intersection of the curve $x^{2}-2xy+2x-2y-17=0$ with the lines $y=0$ and $x+2y+3=0$.

Through the origin, chords $AB$ and $CD$ of the circle $x^{2}+(y-h)^{2}=a^{2}$ are drawn. By considering the pencil of second-order curves passing through the endpoints of the chords, prove that the lines $AD$ and $CB$, and likewise $BD$ and $AC$, meet the axis $Ox$ at equal distances from the origin.

On a second-order curve, six points are taken: 1, 2, 3, 4, 5, 6. The equations of the sides of the resulting hexagon are denoted respectively as follows:

Under these conditions, and for any choice of the coefficients $\lambda$ and $\mu$, the third-order curve $\lambda s_{12}s_{34}s_{56}+\mu s_{23}s_{45}s_{61}=0$ passes through the 6 given points and through the 3 points of intersection of the pairs of opposite sides.

Starting from this, prove Pascal's theorem: in a hexagon inscribed in a second-order curve, the three points of intersection of the pairs of opposite sides lie on one line.