Tangents to conic sections

$\frac{x^{2}}{12} + \frac{y^{2}}{4} = 1$ at the point $(3, 1)$

$\frac{x^{2}}{36} + \frac{y^{2}}{12} = 1$ at the point $(3, - 3)$

$\frac{x^{2}}{9} - \frac{y^{2}}{12} = 1$ at the point $(- 3, 0)$

$\frac{x^{2}}{32} - \frac{y^{2}}{8} = 1$ at the point $(6, 1)$

$xy = 8$ at the point $(4, 2)$

$y^{2} = 6x$ at the point $(3 / 2, 3)$.

$\frac{(x - \alpha )^{2}}{a^{2}} + \frac{(y - \beta )^{2}}{b^{2}} = 1$

$\frac{(x - \alpha )^{2}}{a^{2}} - \frac{(y - \beta )^{2}}{b^{2}} = 1$

$xy = k$

$(y - \beta )^{2} = 2p(x - \alpha )$ at the point $\left(x_{0}, y_{0}\right)$ belonging to the given curve.

the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$;

the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = - 1$;

the hyperbola $xy = k$

the parabola $y^{2} = 2px?$

$3x - 2y - 24 = 0, \frac{x^{2}}{48} + \frac{y^{2}}{36} = 1$

$3x - y - 12 = 0, \frac{x^{2}}{20} - \frac{y^{2}}{36} = 1$

$3x - 16y + 24 = 0, \quad xy = - 3$

$x + y + 1 = 0, y^{2} = 4x$.

parallel to the line $2x - y - 1 = 0$

perpendicular to this same line

making an angle of $45^{\circ }$ with the line $x + 3y + 3 = 0$.

$4x = 3y$

$x = 1$

$x - 2y + 1 = 0$.

$2x + y - 4 = 0$

$y = 3$

$x = 0$.

$\frac{27}{28} x^{2} + \frac{9}{7} y^{2} = 1, 3x + 4y + 5 = 0$

$\frac{27}{28} x^{2} + \frac{9}{7} y^{2} = 1, 3x + 4y = 0$

$6x^{2} - 5y^{2} = 19, 12x + 5y - 6 = 0$

$6x^{2} - 5y^{2} = 19, 12x + 5y = 0$

$y^{2} = 64x, 4x - 3y - 76 = 0$.

from the foci of the ellipse to the tangent line to it at the point $A(1 / 3, 2 / 3)$

between the tangent lines to the ellipse parallel to the line $x + y = 1$.

contains the point $A(- 3, 2)$ and is tangent to the line $4x - 6y - 25 = 0$

is tangent to the lines $x + y - 5 = 0$ and $x + 4y - 10 = 0$.

contains the point $A(4, - 2\sqrt{2})$ and is tangent to the line $3x + y + 8 = 0$;

is tangent to the lines $x = 1$ and $5x - 2y + 3 = 0$.

symmetric with respect to the axis $Oy$ and tangent to the lines $y + 2x = 0, \quad 8x - 2y - 3 = 0$

given by a canonical equation and tangent to the line $x + y + 1 = 0$.

the segment of a tangent line to a hyperbola between its asymptotes is bisected by the point of tangency

all triangles formed by the asymptotes of a hyperbola and an arbitrary tangent line to it have one and the same area; express this area in terms of the semi-axes of the hyperbola.

one of the vertices of the triangle lies on the axis $Ox$

one of the vertices of the triangle lies on the axis $Oy$.

to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

to the parabola $y^{2} = 2px?$

$(- 6, 0)$

$(2.7, \sqrt{7})$

$\left(- 4, - \frac{2}{3} \sqrt{2}\right)$

$(1, 2)$

$(- 2, 2)$

$(1.6, 0)$

$(4, \sqrt{3})$

$(4, 1)$

$(8, 4)$

$(0, 0)$

$(1, - 2)$

$(1, 4)$

$(1, 5)$

$\frac{xx_{0}}{a^{2}} + \frac{yy_{0}}{b^{2}} = 1$ for the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

$\frac{xx_{0}}{a^{2}} - \frac{yy_{0}}{b^{2}} = 1$ for the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

$yy_{0} = p\left(x + x_{0}\right)$ for the parabola $y^{2} = 2px$.

$\frac{x^{2}}{20} + \frac{y^{2}}{5} = 1$ and $\frac{x^{2}}{80} + \frac{4y^{2}}{5} = 1$

$\frac{x^{2}}{5} - \frac{y^{2}}{4} = 1$ and $\frac{x^{2}}{4} - \frac{y^{2}}{3} = 1$

$\frac{x^{2}}{3} + y^{2} = 1$ and $y^{2} = 2x$

$\frac{x^{2}}{3} - y^{2} = 1$ and $y^{2} = 2x$

$y^{2} = 2x$ and $\frac{x^{2}}{3} - \frac{y^{2}}{2} = - 1$

$\frac{x^{2}}{6} + \frac{y^{2}}{3} = 1$ and $\frac{x^{2}}{25} - \frac{y^{2}}{16} = 1$

$y^{2} = \frac{8}{9} x$ and $y^{2} = x - 1$.

the normal to an ellipse at an arbitrary point of it bisects the angle formed by the rays issuing from this point and passing through the foci of the ellipse

the tangent line to a hyperbola at an arbitrary point of it bisects the angle formed by the rays issuing from this point and passing through the foci of the hyperbola

the normal to a parabola at an arbitrary point of it bisects the angle formed by the ray issuing from this point and passing through the focus of the parabola, and the ray issuing from this point, lying inside the parabola and parallel to its axis.

the tangent lines at the points of intersection of an ellipse and a hyperbola having common foci are mutually perpendicular

the tangent lines at the points of intersection of two parabolas with a common focus and oppositely directed axes are mutually perpendicular.

parallel to the line $6x + 17y - 4 = 0$

perpendicular to the line $41x - 24y + 3 = 0$

parallel to the line $y = 2$.

$(3, 3)$

$(0, -0.8)$

$(0, 1)$.