抛物线弦长试题和答案

抛物线弦长试题和答案

一、单项选择题(每题2分,共10题)1.抛物线\(y^2=4x\)过焦点的弦长为\(4\),该弦所在直线斜率为（）A.\(1\)B.\(\pm1\)C.\(2\)D.\(\pm2\)2.抛物线\(x^2=8y\)上一点到焦点的距离为\(4\),则该点到\(x\)轴距离为()A.\(1\)B.\(2\)C.\(3\)D.\(4\)3.抛物线\(y^2=2px(p\gt0)\)过焦点弦两端点坐标为\((x_1,y_1)\),\((x_2,y_2)\),则\(y_1y_2\)的值为()A.\(p^2\)B.\(-p^2\)C.\(2p^2\)D.\(-2p^2\)4.抛物线\(y^2=6x\)的焦点弦长为\(9\),则弦所在直线倾斜角为()A.\(\frac{\pi}{6}\)或\(\frac{5\pi}{6}\)B.\(\frac{\pi}{4}\)或\(\frac{3\pi}{4}\)C.\(\frac{\pi}{3}\)或\(\frac{2\pi}{3}\)D.\(\frac{\pi}{2}\)5.过抛物线\(y^2=4x\)焦点\(F\)的直线交抛物线于\(A\)、\(B\)两点,\(|AF|=3\),则\(|BF|\)为()A.\(1\)B.\(\frac{3}{2}\)C.\(2\)D.\(3\)6.抛物线\(y=\frac{1}{4}x^2\)的焦点弦长为\(5\),则弦的端点纵坐标之和为()A.\(3\)B.\(4\)C.\(5\)D.\(6\)7.抛物线\(y^2=12x\)过焦点弦的中点横坐标为\(4\),则弦长为()A.\(10\)B.\(11\)C.\(12\)D.\(13\)8.过抛物线\(y^2=2x\)焦点的直线与抛物线交于\(A\)、\(B\)两点,\(|AB|=4\),则\(AB\)中点到\(y\)轴距离为()A.\(\frac{3}{2}\)B.\(1\)C.\(\frac{1}{2}\)D.\(2\)9.抛物线\(x^2=-8y\)过焦点弦长为\(10\),则弦所在直线斜率为()A.\(\pm\frac{1}{2}\)B.\(\pm1\)C.\(\pm2\)D.\(\pm3\)10.抛物线\(y^2=16x\)的焦点弦与抛物线交于\(A\)、\(B\)两点,若\(|AF|=10\),则\(|BF|\)为()A.\(6\)B.\(\frac{16}{3}\)C.\(\frac{20}{3}\)D.\(8\)二、多项选择题(每题2分,共10题)1.对于抛物线\(y^2=2px(p\gt0)\)的焦点弦\(AB\),以下说法正确的是（）A.弦长\(|AB|=x_1+x_2+p\)B.若\(A(x_1,y_1)\),\(B(x_2,y_2)\),则\(y_1y_2=-p^2\)C.弦\(AB\)最短时,其所在直线垂直于\(x\)轴D.弦\(AB\)中点到准线距离等于弦长的一半2.已知抛物线\(x^2=4y\),过焦点的弦\(AB\),则()A.弦长\(|AB|\)最小值为\(4\)B.若\(A(x_1,y_1)\),\(B(x_2,y_2)\),则\(x_1x_2=-4\)C.弦\(AB\)所在直线斜率可以为\(0\)D.弦\(AB\)中点纵坐标的最小值为\(2\)3.抛物线\(y^2=8x\),过焦点的弦\(AB\)满足()A.焦点到弦\(AB\)中点的距离最小值为\(2\)B.若\(|AB|=10\),则弦\(AB\)中点横坐标为\(3\)C.弦\(AB\)所在直线倾斜角为\(\frac{\pi}{4}\)时,弦长为\(16\)D.弦\(AB\)两端点到准线距离之和为\(|AB|\)4.对于抛物线\(y^2=6x\)的焦点弦\(AB\),下列结论正确的是()A.焦点\(F\)坐标为\((\frac{3}{2},0)\)B.若\(|AB|=9\),则弦\(AB\)中点横坐标为\(3\)C.弦\(AB\)最短时长度为\(6\)D.弦\(AB\)所在直线斜率不存在时,弦长为\(6\)5.抛物线\(x^2=-10y\)过焦点弦\(AB\),以下说法正确的是（）A.焦点坐标为\((0,-\frac{5}{2})\)B.若弦\(AB\)中点纵坐标为\(-\frac{5}{2}\),则弦长为\(10\)C.弦\(AB\)所在直线斜率可以不存在D.弦\(AB\)中点到\(x\)轴距离最小值为\(\frac{5}{2}\)6.已知抛物线\(y^2=2x\),过焦点的弦\(AB\),则()A.焦点坐标为\((\frac{1}{2},0)\)B.若\(|AB|=3\),则弦\(AB\)中点横坐标为\(1\)C.弦\(AB\)所在直线倾斜角为\(\frac{\pi}{3}\)时,弦长为\(\frac{8}{3}\)D.弦\(AB\)两端点到\(y\)轴距离之和为\(|AB|-1\)7.抛物线\(y^2=10x\)的焦点弦\(AB\),以下正确的是()A.焦点坐标为\((\frac{5}{2},0)\)B.若\(|AB|=15\),则弦\(AB\)中点横坐标为\(5\)C.弦\(AB\)所在直线斜率为\(1\)时,弦长为\(20\)D.弦\(AB\)中点到准线距离为\(\frac{|AB|}{2}\)8.对于抛物线\(x^2=6y\)过焦点的弦\(AB\),说法正确的是()A.焦点坐标为\((0,\frac{3}{2})\)B.若弦\(AB\)中点纵坐标为\(3\),则弦长为\(9\)C.弦\(AB\)所在直线斜率可以为任意实数D.弦\(AB\)中点到\(x\)轴距离最小值为\(\frac{3}{2}\)9.抛物线\(y^2=18x\)过焦点弦\(AB\),以下结论正确的是()A.焦点坐标为\((\frac{9}{2},0)\)B.若\(|AB|=27\),则弦\(AB\)中点横坐标为\(9\)C.弦\(AB\)所在直线倾斜角为\(\frac{\pi}{6}\)时,弦长为\(36\)D.弦\(AB\)两端点到准线距离之差的绝对值为\(|AB|\)10.已知抛物线\(x^2=-12y\)过焦点弦\(AB\),则()A.焦点坐标为\((0,-3)\)B.若弦\(AB\)中点纵坐标为\(-6\),则弦长为\(18\)C.弦\(AB\)所在直线斜率不存在时,弦长为\(12\)D.弦\(AB\)中点到\(x\)轴距离最小值为\(3\)三、判断题(每题2分,共10题)1.抛物线\(y^2=4x\)过焦点弦长一定大于\(4\)。()2.抛物线\(x^2=2py(p\gt0)\)过焦点弦两端点纵坐标之积为\(-p^2\)。()3.抛物线\(y^2=8x\)焦点弦最短长度是\(8\)。()4.过抛物线\(y^2=2px(p\gt0)\)焦点的直线与抛物线交于\(A\)、\(B\)两点,若\(|AF|=|BF|\),则弦\(AB\)垂直于\(x\)轴。（)5.抛物线\(x^2=-4y\)过焦点弦长最小值为\(4\)。()6.抛物线\(y^2=6x\)焦点弦中点到准线距离等于弦长的一半。()7.过抛物线\(y^2=10x\)焦点的直线倾斜角为\(\frac{\pi}{4}\)时,弦长为\(20\)。（)8.抛物线\(x^2=8y\)过焦点弦两端点横坐标之积为\(-16\)。()9.抛物线\(y^2=12x\)焦点弦所在直线斜率不存在时,弦长为\(12\)。（)10.抛物线\(x^2=-10y\)过焦点弦中点到\(x\)轴距离最小值为\(\frac{5}{2}\)。(）四、简答题(每题5分,共4题)1.求抛物线\(y^2=4x\)过焦点且斜率为\(1\)的弦长。答案:抛物线\(y^2=4x\)焦点为\((1,0)\),直线方程为\(y=x-1\),联立抛物线方程得\((x-1)^2=4x\),即\(x^2-6x+1=0\),设两根为\(x_1\),\(x_2\),\(x_1+x_2=6\),弦长\(=x_1+x_2+p=6+2=8\)。2.已知抛物线\(x^2=8y\),过焦点的弦\(AB\)中点纵坐标为\(6\),求弦长。答案:抛物线\(x^2=8y\)焦点为\((0,2)\),准线\(y=-2\)。设\(A(x_1,y_1)\),\(B(x_2,y_2)\),由中点纵坐标为\(6\),得\(y_1+y_2=12\),弦长\(=y_1+y_2+p=12+4=16\)。3.抛物线\(y^2=16x\),过焦点弦所在直线倾斜角为\(\frac{\pi}{3}\),求弦长。答案:抛物线\(y^2=16x\)焦点\((4,0)\),直线斜率\(\sqrt{3}\),直线方程\(y=\sqrt{3}(x-4)\),联立方程得\(3(x-4)^2=16x\),整理得\(3x^2-40x+48=0\),设两根\(x_1\),\(x_2\),\(x_1+x_2=\frac{40}{3}\),弦长\(=x_1+x_2+p=\frac{40}{3}+8=\frac{64}{3}\)。4.抛物线\(x^2=-12y\),过焦点弦长为\(16\),求弦中点纵坐标。答案：抛物线\(x^2=-12y\)焦点\((0,-3)\),准线\(y=3\)。设弦端点\(A(x_1,y_1)\),\(B(x_2,y_2)\),弦长\(=-(y_1+y_2)+p=16\),即\(-(y_1+y_2)+6=16\),得\(y_1+y_2=-10\),中点纵坐标为\(-5\)。五、讨论题(每题5分,共4题)1.讨论抛物线\(y^2=2px(p\gt0)\)焦点弦长与直线斜率的关系。答案:设焦点弦所在直线斜率为\(k\)(\(k\neq0\)时),直线方程\(y=k(x-\frac{p}{2})\),与抛物线联立得\(k^2(x-\frac{p}{2})^2=2px\),利用韦达定理得\(x_1+x_2\),弦长\(=x_1+x_2+p\)。当\(k=0\)时弦长为\(p\);\(k\)不存在时,弦长为\(2p\)。斜率绝对值越大,弦长越大。2.如何求抛物线\(x^2=2py(p\gt0)\)过焦点弦中点的轨迹方程?答案:设弦端点\(A(x_1,y_1)\),\(B(x_2,y_2)\),焦点\((0,\frac{p}{2})\),弦所在直线方程\(y=kx+\frac{p}{2}\),联立抛物线方程得\(x^2-2pkx-p^2=0\),\(x_1+x_2=2pk\),\(y_1+y_2=k(x_1+x_2)+p=2pk^2+p\),中点坐标\((\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\),消去\(k\)得\(x^2=p(y-\frac{p}{2})\)。3.抛物线焦点弦有哪些重要性质,举例说明其应用