Exercises for Seminar 4 (curves of the second order)

Date: 2015-10-07; view: 461.

4.1. Find radius and the coordinates of center of circumference:

a) b)

4.2. Find lengths of semi-axes, eccentricity, the coordinates of focuses, compose equations of directrices of an ellipse: a) b) .

4.3. Let an ellipse be given. Determine whether the point lies on the ellipse, inside or outside of it.

4.4. An ellipse in a given system of coordinates has a canonic equation. Compose this equation if:

a) the distance between vertices lying on big axis is 16, and the distance between focuses is 10.

b) focuses of the ellipse are the points , and the point belongs to the ellipse.

4.5. Find semi-axes, eccentricity, the coordinates of focuses, compose equations of directrices and asymptotes of the following hyperbola: a) b)

4.6. A hyperbola is given. Determine whether the point lies on the hyperbola, inside of one of its branches or between branches: a) b) .

4.7. A hyperbola in a given system of coordinates has a canonic equation. Compose this equation if:

a) the distance between vertices is 10, and the distance between focuses is 12.

b) directrices of the hyperbola are lines , and the point (– 9, 4) belongs to the hyperbola.

4.8. Find the coordinates of focus and compose the equation of directrix of the following parabola:

a) b) .

4.9. A parabola in a given system of coordinates has a canonic equation. Compose this equation if the point belongs to the parabola.

4.10. Determine the type of the following curve of the second order, compose its canonic equation and find the canonic system of coordinates:

a) ; b) .