英文的数学题Problem1Considerthequadraticequationx2+y2+Dx+Ey+F=0:WhatconditiononthevaluesofD,EandFdetermineswhetherthisacircle?Usethisconditiontoshowthatx2+y2¡2x¡4y+3=0isacircleandsketch

Question

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.)
不死翻译

Answer 1

问题1考虑二次方程x2 + y2 + + dx的安永+六= 0 :什么条件就值d ,e和f确定是否这是一个大圈?利用这一条件,以表明方程x2 + y2倍速¡ ¡ 4y + 3 = 0 ,是一个圈 素描圈.请问点( 4,6 ; 3 )所在内,外或本圈?给一个理由,你的答案.
问题2寻找标准代数形式的方程,椭圆的一个焦点( 4 ; 0 ) ,准× = 9和偏心2 / 3 .还有椭圆与这些财产?若有,ˆ次方程.素描椭圆(美国) .
问题3地球绕太阳的椭圆与太阳的一个重点.长轴经典£ 108英里和偏心是0.0167 .找到最小和最大距离地球从太阳.
问题4椭圆台球桌兴建的口袋.一个球是放置在一个焦点,并击中玩家的提示,在任何方向.显示皮球穿过其他重点之后,一个反弹.(提示:考虑下面的图表,而线l是切向椭圆点页显示,截至 问沿着l时,jf1qj + jf2qj是最低时问=页,让f01应重新° 13765700 f1在草显示的f2 ,磷 和f01躺在一条直线) .
别太懒,自己做吧!