Determine whether the given equation is an equation of the conic section. If so, identify the type of a conic section and its properties (the vertex, the center, the radius, the semi-major and semi-minor axis, the eccentricity) :

Determine the relative position of a straight line *p* and a circle *k*. If they have any intersection points, determine their coordinates :

Find the equation of a circle circumscribed about the triangle *ABC*, where *A* [3;1], *B* [2;–2], *C* [6;6].

Find the equation of a circle passing through the points *K* [2;6], *L* [6;2], if the center of a circle lies on the straight line *p*: 2*x* + 3*y* – 5 = 0.

Find the equation of an ellipse having the center at the origin of the coordinate system and passing through the points *M* [2;] and *N* [6;0].

Find the vertex equation of a parabola passing through the points *A* [3;3], *B* [0;12] and *C* [4;6], if the axis of a parabola is parallel to the *x-*axis.

Find the equation of a hyperbola passing through the point *H* [4;9], if the asymptotes of a hyperbola are given by the equations *y* – 3 = ± 2( *x* + 1 ).

Find the equation of an ellipse, if two of the vertices of an ellipse have coordinates *C* [3;7], *D* [–5;7] and the focus of an ellipse has coordinates *F* [–1;4].

Find the vertex equation of a parabola passing through the point *P* [4;–5], if the vertex of a parabola has coordinates *V* [3;–7].

Find the equation of a hyperbola if the axis of a hyperbola is parallel to the *x-*axis, the center of a hyperbola is *C* [1;–1], the semi-major axis of a hyperbola is *a* = and the eccentricity of a hyperbola is *e* = .

Find the equation of a circle whose diameter is a line segment *AB*, *A* [2;–5], *B* [–4;1].

Find the equation of an ellipse, if two of the vertices of an ellipse have coordinates *A* [0;–3], *B* [0;3] and the distance between foci of an ellipse is 8.

Find the equations of tangent lines to the circle *x*^{2} + *y*^{2} = 25 that pass through the point *A* [7;1].

Find the equations of tangent lines to the ellipse 9*x*^{2} + 25*y*^{2} – 18*x* + 100*y* – 116 = 0 that pass through the point *B* [–4;7].

Find the equations of tangent lines to the parabola *y*^{2} = 8*x* that pass through the point *C* [–3;1].

Find the equations of tangent lines to the hyperbola *x*^{2} – 9*y*^{2} = 25 that pass through the point *D* [–5;–5/3].

Find the equation of a tangent line to the circle *x*^{2} + *y*^{2} – 6*x* – 4*y* – 3 = 0 that is perpendicular to a straight line *p*: 4*x* + *y* – 9 = 0.

Find the equation of a tangent line to the ellipse 9*x*^{2} + 16*y*^{2} = 144 with the slope equal to the value of *k* = 1.

Find the equation of a tangent line to the parabola *y*^{2} – 6*x* – 6*y* + 3 = 0 that is parallel to a straight line *p*: 3*x* – 2*y* + 7 = 0.

Find the equation of a tangent line to the hyperbola 4*x*^{2} – *y*^{2} = 36 that is parallel to a straight line *p*: 5*x* – 2*y* + 7 = 0.

Find the intersection points of the hyperbola 4( *x* – 4 )^{2} – ( *y* – 2 )^{2} = 16 and the circle

( *x* – 4 )^{2} + ( *y* – 2 )^{2} = 64.

Determine the relative position of two circles ( *x* – 3 )^{2} + *y*^{2} = 45 and ( *x* – 9 )^{2} + ( *y* – 2 )^{2} = 25.

Find the equation of a sphere having its center in *C* [2;0;–3], if a sphere touches the plane*ρ*: *x* + *y* – 3 = 0.

Find the center and the radius of a sphere *ω*: *x*^{2} + *y*^{2} + *z*^{2} + 12*x* – 14*y* + 16*z* – 100 = 0.

Determine the coordinates of points of intersection of a straight line*p*: {*x* = 1 – *t*, *y* = 3 + *t*, *z* = 2 + *t*, *t*∈**R**} and a sphere *τ *: ( *x* – 1 )^{2} + *y*^{2} + ( *z* – 2 )^{2} = 9.

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