Exercises

Conics and quadrics

Problems to solve on your own. The final solution appears with the button below each problem; each exercise also has its own page, to share with the link.

1. Classifying a conic with a mixed term

Classify the conic

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

Solution available to those who have unlocked the subject.

2. Classifying an ellipse

Classify the conic

x^{2} + 2 x y + 2 y^{2} - 2 x - 2 y - 1 = 0.

Solution available to those who have unlocked the subject.

3. Canonical form of a hyperbola

Find the center and the canonical form of the conic

7 x^{2} + 6 x y - y^{2} - 14 x - 6 y - 1 = 0.

Solution available to those who have unlocked the subject.

4. Canonical form of an ellipse

Find the center, the canonical form and the semi-axes of the conic

6 x^{2} + 4 x y + 9 y^{2} - 8 x + 14 y + 1 = 0.

Solution available to those who have unlocked the subject.

5. Center and asymptotes of a hyperbola

Verify that the conic

x^{2} - 2 x y - 3 y^{2} + 2 x + 6 y - 7 = 0

is a hyperbola and find its center and asymptotes.

Solution available to those who have unlocked the subject.

6. Axis and vertex of a parabola

Find the axis, the vertex and the canonical form of the parabola

4 x^{2} - 4 x y + y^{2} - 14 x - 18 y + 31 = 0.

Solution available to those who have unlocked the subject.

7. Vertex and parameter of a parabola

Find the axis, the vertex and the canonical form of the parabola

x^{2} + 2 x y + y^{2} - 12 x + 4 y + 20 = 0.

Solution available to those who have unlocked the subject.

8. Conic with a parameter in the coefficient of y²

Classify, as $k \in R$ varies, the conic

x^{2} + 4 x y + k y^{2} + 2 x = 0.

Solution available to those who have unlocked the subject.

9. Conic with a parameter and parallel lines

Classify, as $k \in R$ varies, the conic

x^{2} + 2 x y + k y^{2} + 2 x + 2 y = 0.

Solution available to those who have unlocked the subject.

10. Degenerate conic consisting of parallel lines

Verify that the conic

x^{2} - 4 x y + 4 y^{2} + 2 x - 4 y - 3 = 0

is degenerate and write it as the union of two lines.

Solution available to those who have unlocked the subject.

11. Recognizing a hyperboloid

Recognize the quadric

x^{2} + y^{2} - 2 z^{2} - 2 x + 4 y + 8 z - 5 = 0

and find its center and canonical form.

Solution available to those who have unlocked the subject.

12. Recognizing a paraboloid

Recognize the quadric

x^{2} + 2 x y + 2 y^{2} - 2 z = 0.

Solution available to those who have unlocked the subject.

13. Quadric with a parameter

Recognize, as $k \in R$ varies, the quadric

x^{2} + y^{2} + k z^{2} + 2 z = 0.

Solution available to those who have unlocked the subject.