一道高中数学圆的题目若c≠0,则直线ax + by + c = 0与圆 x^2 + y^2 + ax + by + c = 0的交点个数是_______.给出过程,谢谢!
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![一道高中数学圆的题目若c≠0,则直线ax + by + c = 0与圆 x^2 + y^2 + ax + by + c = 0的交点个数是_______.给出过程,谢谢!](/uploads/image/z/7650598-22-8.jpg?t=%E4%B8%80%E9%81%93%E9%AB%98%E4%B8%AD%E6%95%B0%E5%AD%A6%E5%9C%86%E7%9A%84%E9%A2%98%E7%9B%AE%E8%8B%A5c%E2%89%A00%2C%E5%88%99%E7%9B%B4%E7%BA%BFax+%2B+by+%2B+c+%3D+0%E4%B8%8E%E5%9C%86+x%5E2+%2B+y%5E2+%2B+ax+%2B+by+%2B+c+%3D+0%E7%9A%84%E4%BA%A4%E7%82%B9%E4%B8%AA%E6%95%B0%E6%98%AF_______.%E7%BB%99%E5%87%BA%E8%BF%87%E7%A8%8B%2C%E8%B0%A2%E8%B0%A2%21)
一道高中数学圆的题目若c≠0,则直线ax + by + c = 0与圆 x^2 + y^2 + ax + by + c = 0的交点个数是_______.给出过程,谢谢!
一道高中数学圆的题目
若c≠0,则直线ax + by + c = 0与圆 x^2 + y^2 + ax + by + c = 0的交点个数是_______.
给出过程,谢谢!
一道高中数学圆的题目若c≠0,则直线ax + by + c = 0与圆 x^2 + y^2 + ax + by + c = 0的交点个数是_______.给出过程,谢谢!
假设有交点则:
ax + by + c = 0代入x^2 + y^2 + ax + by + c = 0
得x^2 + y^2 =0
得x=y=0代入ax + by + c=0得c=0与c≠0矛盾
所以无交点
圆心为(-a/2,-b/2) 半径为 r=(a^2/4+b^2/4-c)^1/2
圆心到直线距离为 d=|-a^2/2-b^2/2+c|/(a^2+b^2)^1/2
d^2/r^2=[(a^2+b^2)^2-4(a^2+b^2)+4c^2]/[(a^2+b^2)^2-4(a^2+b^2)]>1
d>r
所以无交点
由ax + by + c = 0得y=-(ax+c)/b 代入x^2 + y^2 + ax + by + c = 0
得x^2 + y^2 =0
∴x=y=0
将x=y=0代入ax + by + c=0得c=0
∴与c≠0矛盾
∴无交点
看圆心距
联立方程ax+by+c=0和x^2+y^2+ax+by+c=0 得x^2+y^2=0,解出来x=0,y=0. 再将x=0,y=0代入ax+by+c=0,得c=0与c≠0矛盾。故没有交点
要讨论c的值的,是填空或选择吗?是问答吧?c只确定后相切。是个推论!