COORDINATE GEOMETRY III (CONICS) ASSIGNMENT

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: 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 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 x² + y² = 25 that pass through the point A [7;1].

Find the equations of tangent lines to the ellipse 9x² + 25y² – 18x + 100y – 116 = 0 that pass through the point B [–4;7].

Find the equations of tangent lines to the parabola y² = 8x that pass through the point C [–3;1].

Find the equations of tangent lines to the hyperbola x² – 9y² = 25 that pass through the point D [–5;–5/3].

Find the equation of a tangent line to the circle x² + y² – 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 9x² + 16y² = 144 with the slope equal to the value of k = 1.

Find the equation of a tangent line to the parabola y² – 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 4x² – y² = 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
( x – 4 )² + ( y – 2 )² = 64.

Determine the relative position of two circles ( x – 3 )² + y² = 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: x² + y² + z² + 12x – 14y + 16z – 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 + y² + ( z – 2 )² = 9.