24.解:(1)解法1:根据题意可得:A(-1,0),B(3,0);

则设抛物线的解析式为(a≠0)

又点D(0,-3)在抛物线上,∴a(0+1)(0-3)=-3,解之得:a=1

 ∴y=x2-2x-3····································································································· 3分

自变量范围:-1≤x≤3···················································································· 4分

      解法2:设抛物线的解析式为(a≠0)

       根据题意可知,A(-1,0),B(3,0),D(0,-3)三点都在抛物线上

,解之得:

y=x2-2x-3····································································································· 3分

自变量范围:-1≤x≤3····························································· 4分

      (2)设经过点C“蛋圆”的切线CEx轴于点E,连结CM

       在RtMOC中,∵OM=1,CM=2,∴∠CMO=60°,OC=

       在RtMCE中,∵OC=2,∠CMO=60°,∴ME=4

 ∴点CE的坐标分别为(0,),(-3,0) ·················································· 6分

∴切线CE的解析式为··························································· 8分

 (3)设过点D(0,-3),“蛋圆”切线的解析式为:y=kx-3(k≠0) ························· 9分

        由题意可知方程组只有一组解

  即有两个相等实根,∴k=-2············································· 11分

  ∴过点D“蛋圆”切线的解析式y=-2x-3····················································· 12分

 

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