11 Questions on Ellipses, Hyperbola, Parabola

1 The equation xsquared + ysquared =1 represents an ellipse.
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a) State the lengths of the major and the minor axes.

b) State the x-intercepts and y-intercepts.

c) Find the coordinates of loci.

d) Find the points of intersection with the line y=1-2x.

e) Graph the ellipse, line, and points of intersection.

f) Find the equation of the ellipse after it is translated according to ((x,y) arrow (x-3, y+1).

2. An ellipse is represented by the equation xsquared+4ysquared-4x+8y-60=0.

a) Express the equation of the ellipse in standard form.

b) Find the centre of the ellipse.

c) State the lengths of the major and minor axes.

d) Find the coordinates of the foci.

e) Graph the ellipse, showing the centre and the foci.

3. An elliptical pool table is 5m long at itslongest point and 3m wide at its widest point. The pool table has two holes at the positions of the foci.

a) Determine the equation of the ellipse.

b) Determine the positions of the foci.

c) Sketch the pool table, showing the positions of the foci.

d) If a pool ball is hit from one focus, it will bounce once and enter the hole at the other focus. Will the ball always travel the same distance if it is hit from one focus? If so, what is the distance it will travel? Prove the statement by selecting any point (except one of the verticals) on the edge of the pool table.

4. A model of a planet shows a satellite moving in an elliptical orbit about the planet. The closest distance from the centre of the satellite to the centre of the planet in the model is 3 m and the longest distance is 8m.

a) Determine a possible equation for the orbit of the satellite.

b) Sketch the orbit of the satellite showing the position of the planet.

5. State the key features (vertex, focus, directrix, direction of opening, and axis of symmetry) of each parabola, and sketch the graph.

a) xsquared=20y

b) ysquared=-6x

c) (x-3)squared=-8(y+3)

d) ysquared-2x+4y-7=0

6. Find the equation of each parabola described below.

a) parabola with vertex (0,0) and the focus (0,7)

b) parabola with focus (-3,0) and directrix x=3

c) parabola with vertex (3,3) and directrix x=-1

d) parabola with focus (-2,-1) and directrix y=5

e) ysquared=-6x translated according to ((x,y) arrow(x-2,y+4))

7. A parabolic reflector has a cross section with formula ysquared=0.6x. (Measurements are in metres.)

a) Determine the distance of the focus from the vertex.

b) The diamter of the disk at its widest point is 1.6m. Determine the depth of the reflector.

c) Sktech the reflector, showing the focus and the width at the opening.

8. A parabolic arch has the equation xsquared+10y-10=0. The arch is on a hill with equation y=0.1x-1. (Measurements are in metres.)

a) Find the points of interseciont for the arch and the hill.

b) Sketch the arch and th hill, showing the points of intersection.

c) Find the height of the arch above the hill at the point where x=1. Show this height on your graph.

9. A hyperbola is given by the equation xsquared-ysquared=1.
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a) State the lengths of the transverse axis and conjugate axis.

b) State any x-intercepts or y-intercepts.

c) Find the coordinates of the foci and the equations of the asymptotes.

d) Find the points of intersection with the line y=4-x.

e) Graph the hyperbola, the line, and the points of the asymptotes.

f) Find the equation of the hyperbola after it is translated accoding to ((x,y)arrow(x-3, y+1)).

10. A hyperbola is represented by the equation 4xsquared-ysquared+8x+4y+16=0.

a) Express the equation of the hyperbola in standard form.

b) Find the point about which the hyperbola is centered.

c) Find the coordinates of the foci.

d) Find the equations of the asymptotes.

e) Graph the hyperbola, showing the centre, foci and asymptotes.

11. Two radar stations, situated 100 km apart, detect a moving airplane. The difference in the distances between the airplane and the radar stations is 50km. Assume that the radar stations are on the y-axis centred about the origin.

a) Determine the equation of the hyperbolic path of the airplane.

b) Sketch the graph, showing the locations of the radar stations.