Ellipse practice problems
An ellipse is a conic section that appears as an elongated circle, defined as the set of all points where the sum of distances to two fixed points (foci) is constant. The ellipse has two axes: the major axis (longest diameter) and the minor axis (shortest diameter). Its equation in standard form is (x²/a²) + (y²/b²) = 1, where a and b are the semi-major and semi-minor axes. Ellipses appear in orbital mechanics, as planets orbit the sun in elliptical paths according to Kepler's laws. They also arise in optics, acoustics, and engineering applications. Students learn to identify ellipse parameters, sketch graphs, and solve application problems.Direction: Solve each problem carefully and show your solution in each item.
Number of problems found: 7
- Satellite speed orbit
Determine the average speed and orbit of the Earth's first artificial satellite. Its distance from the Earth's surface in the perigee was 226 km, and in the apogee, 947 km. - Coordinates - ellipse
Write the equation of the ellipse that passes through the points, and its axes are identical to the coordinate axes when A = [2, 3] and B = [−1, −4]. - Tangents to ellipse
Find the magnitude of the angle at which the ellipse x² + 5 y² = 5 is visible from the point P[5, 1]. - Complex plane mapping
Show that the mapping w = z +c/z, where z = x+iy, w = u+iv, and c is a real number, maps the circle |z| = 1 in the z-plane into an ellipse in the (u, v) plane. - The tangent line Find the tangent line of the ellipse 9x² + 16y² = 144 with slope k = -1.
- Ellipse
Ellipse is expressed by equation 9x² + 25y² - 54x - 100y - 44 = 0. Find the length of primary and secondary axes, eccentricity, and coordinates of the ellipse's center. - Circle in rhombus
An inscribed circle is in the rhombus. Contact points of touch divide the sides into parts of length 14 mm and 9 mm. Calculate the circle's area.
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