英文的数学题Problem 1Consider the quadratic equationx2 + y2 + Dx +

英文的数学题
Problem 1
Consider the quadratic equation
x2 + y2 + Dx + Ey + F = 0:
What condition on the values of D,E and F determines whether this a circle?
Use this condition to show that
x2 + y2 ¡ 2x ¡ 4y + 3 = 0
is a circle and sketch the circle.
Does the point (0:4; 3) lie inside,outside or on this circle?Give a reason for
your answer.
Problem 2
Find the standard algebraic form of the equation of an ellipse with one focus at
(4; 0),directrix x = 9 and eccentricity 2/3.Is there another ellipse with these
properties?If so,¯nd its equation.Sketch the ellipse(s).
Problem 3
The Earth orbits the Sun elliptically with the Sun as one focus.The major axis
is 1:86 £ 108 miles and the eccentricity is 0.0167.Find the smallest and largest
distances of the Earth from the Sun.
Problem 4
An elliptical snooker table is constructed without pockets.A ball is placed at one
focus and struck by the player's cue in any direction.Show that the ball passes
through the other focus after a single rebound.
(Hint:Consider the diagram below where the line l is tangent to the ellipse
at the point P.Show that as Q moves along l,jF1Qj+jF2Qj is a minimum when
Q = P.Let F01 be the re°ection of F1 in l.Show that F2,P and F01 lie on a
straight line.)
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