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Math Exercises: Analytic Geometry of the Conic Sections

 

 

 

 

1. 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) :

 

Analytic geometry of the conic sections - Exercise 1

 

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

 

Analytic geometry of the conic sections - Exercise 2

 

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

 

4. 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.

 

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

 

6. 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.

 

7. 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 ).

 

8. 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].

 

9. 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].

 

10. 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 = Square root of five and the eccentricity of a hyperbola is e = Square root of seven.

 

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

 

12. 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.

 

13. Find the equations of tangent lines to the circle x2 + y2 = 25 that pass through the point A [7;1].

 

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

 

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

 

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

 

17. 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.

 

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

 

19. 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.

 

20. Find the equation of a tangent line to the hyperbola 4x2y2 = 36 that is parallel to a straight line p: 5x – 2y + 7 = 0.

 

21. 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.

 

22. Determine the relative position of two circles ( x – 3 )2 + y2 = 45 and ( x – 9 )2 + ( y – 2 )2 = 25.

 

23. 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.

 

24. Find the center and the radius of a sphere ω: x2 + y2 + z2 + 12x – 14y + 16z – 100 = 0.

 

25. Determine the coordinates of points of intersection of a straight line
p: {x = 1 – t, y = 3 + t, z = 2 + t, tR} and a sphere τ : ( x – 1 )2 + y2 + ( z – 2 )2 = 9.

 

 

 

 

 

 

You might be also interested in:

- Vectors

- Relative Positions of Lines and Planes

- Straight Lines and Planes

- Matrices

 

 
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