抛物线y^2=2px(p>0)上有A(x1,y1)B(x2,y2)C(x3,y3)三点,F是它的焦点若|AF|,|BF|,|CF|成等差数列,则()A.2 x2 = x1 + x3B.2 y2 = y1 + y3C.2 x3 = x1 + x2D.2 y3 = y1 + y2
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![抛物线y^2=2px(p>0)上有A(x1,y1)B(x2,y2)C(x3,y3)三点,F是它的焦点若|AF|,|BF|,|CF|成等差数列,则()A.2 x2 = x1 + x3B.2 y2 = y1 + y3C.2 x3 = x1 + x2D.2 y3 = y1 + y2](/uploads/image/z/3136079-47-9.jpg?t=%E6%8A%9B%E7%89%A9%E7%BA%BFy%5E2%3D2px%EF%BC%88p%EF%BC%9E0%29%E4%B8%8A%E6%9C%89A%28x1%2Cy1%29B%28x2%2Cy2%29C%28x3%2Cy3%29%E4%B8%89%E7%82%B9%2CF%E6%98%AF%E5%AE%83%E7%9A%84%E7%84%A6%E7%82%B9%E8%8B%A5%7CAF%7C%2C%7CBF%7C%2C%7CCF%7C%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E5%88%99%EF%BC%88%EF%BC%89A.2+x2+%3D+x1+%2B+x3B.2+y2+%3D+y1+%2B+y3C.2+x3+%3D+x1+%2B+x2D.2+y3+%3D+y1+%2B+y2)
抛物线y^2=2px(p>0)上有A(x1,y1)B(x2,y2)C(x3,y3)三点,F是它的焦点若|AF|,|BF|,|CF|成等差数列,则()A.2 x2 = x1 + x3B.2 y2 = y1 + y3C.2 x3 = x1 + x2D.2 y3 = y1 + y2
抛物线y^2=2px(p>0)上有A(x1,y1)B(x2,y2)C(x3,y3)三点,F是它的焦点若|AF|,|BF|,|CF|成等差数列,则()
A.2 x2 = x1 + x3
B.2 y2 = y1 + y3
C.2 x3 = x1 + x2
D.2 y3 = y1 + y2
抛物线y^2=2px(p>0)上有A(x1,y1)B(x2,y2)C(x3,y3)三点,F是它的焦点若|AF|,|BF|,|CF|成等差数列,则()A.2 x2 = x1 + x3B.2 y2 = y1 + y3C.2 x3 = x1 + x2D.2 y3 = y1 + y2
根据抛物线的第二定义可知:|AF|=x1+p/2;|BF|=x2+p/2;|CF|=x3+p/2.
又|AF|,|BF|,|CF|成等差数列,可知|AF|+|CF|=2|BF|,即2 x2 = x1 + x3.
故应选A.
因为A(x1,y1)B(x2,y2)C(x3,y3)三点在抛物线上
且抛物线上的点到焦点的距离等于点到准线的距离
所以
|AF|=x1+ p/2 |BF|=x2+ p/2 |CF|=x3+ p/2
又因为 |AF| , |BF| , |CF| 成等差数列
所以 2*|BF|=|AF|+|CF|
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因为A(x1,y1)B(x2,y2)C(x3,y3)三点在抛物线上
且抛物线上的点到焦点的距离等于点到准线的距离
所以
|AF|=x1+ p/2 |BF|=x2+ p/2 |CF|=x3+ p/2
又因为 |AF| , |BF| , |CF| 成等差数列
所以 2*|BF|=|AF|+|CF|
2*(x2+ p/2)=(x1+ p/2)+(x3+ p/2)
2 * x2 + p=x1 + x3 + p
则 2 x2 =x1+x3
答案选A
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