Math Exercises & Math Problems: Analytic Geometry of the Conic Sections 
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 semimajor and semiminor 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: 2x + 3y – 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 xaxis.
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 xaxis, the center of a hyperbola is C [1;–1], the semimajor 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 9x^{2} + 25y^{2} – 18x + 100y – 116 = 0 that pass through the point B [–4;7].
Find the equations of tangent lines to the parabola y^{2} = 8x that pass through the point C [–3;1].
Find the equations of tangent lines to the hyperbola x^{2} – 9y^{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} – 6x – 4y – 3 = 0 that is perpendicular to a straight line p: 4x + y – 9 = 0.
Find the equation of a tangent line to the ellipse 9x^{2} + 16y^{2} = 144 with the slope equal to the value of k = 1.
Find the equation of a tangent line to the parabola y^{2} – 6x – 6y + 3 = 0 that is parallel to a straight line p: 3x – 2y + 7 = 0.
Find the equation of a tangent line to the hyperbola 4x^{2} – y^{2} = 36 that is parallel to a straight line p: 5x – 2y + 7 = 0.
Find the intersection points of the hyperbola 4( x – 4 )^{2} – ( y – 2 )^{2} = 16 and the circle
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
Find the center and the radius of a sphere ω: x^{2} + y^{2} + z^{2} + 12x – 14y + 16z – 100 = 0.
Determine the coordinates of points of intersection of a straight line
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