全國中學(xué)生數(shù)學(xué)競賽解析幾何試題及答案1.已知點(diǎn)\(A(1,2)\)，\(B(3,4)\)，則線段\(AB\)的中點(diǎn)坐標(biāo)為（）A.\((2,3)\)B.\((3,2)\)C.\((4,6)\)D.\((6,4)\)答案：A2.直線\(y=2x+1\)的斜率為（）A.1B.2C.3D.4答案：B3.圓\(x^2+y^2=4\)的半徑為（）A.1B.2C.3D.4答案：B4.拋物線\(y=x^2\)的焦點(diǎn)坐標(biāo)為（）A.\((0,\frac{1}{4})\)B.\((0,\frac{1}{2})\)C.\((\frac{1}{4},0)\)D.\((\frac{1}{2},0)\)答案：A5.橢圓\(\frac{x^2}{9}+\frac{y^2}{4}=1\)的長軸長為（）A.2B.3C.4D.6答案：D6.雙曲線\(\frac{x^2}{4}-\frac{y^2}{9}=1\)的漸近線方程為（）A.\(y=\pm\frac{3}{2}x\)B.\(y=\pm\frac{2}{3}x\)C.\(y=\pm\frac{4}{9}x\)D.\(y=\pm\frac{9}{4}x\)答案：A7.已知直線\(l\)過點(diǎn)\((1,1)\)且與直線\(y=-2x+3\)垂直，則直線\(l\)的方程為（）A.\(y=\frac{1}{2}x+\frac{1}{2}\)B.\(y=\frac{1}{2}x-\frac{1}{2}\)C.\(y=2x-1\)D.\(y=2x+1\)答案：A8.圓\((x-1)^2+(y-2)^2=9\)的圓心坐標(biāo)為（）A.\((1,2)\)B.\((-1,-2)\)C.\((2,1)\)D.\((-2,-1)\)答案：A9.拋物線\(y=-x^2\)的準(zhǔn)線方程為（）A.\(y=\frac{1}{4}\)B.\(y=-\frac{1}{4}\)C.\(x=\frac{1}{4}\)D.\(x=-\frac{1}{4}\)答案：A10.橢圓\(\frac{x^2}{16}+\frac{y^2}{25}=1\)的離心率為（）A.\(\frac{3}{5}\)B.\(\frac{4}{5}\)C.\(\frac{9}{25}\)D.\(\frac{16}{25}\)答案：A11.雙曲線\(\frac{x^2}{16}-\frac{y^2}{9}=1\)的實(shí)軸長為（）A.2B.4C.6D.8答案：D12.已知點(diǎn)\(P(x,y)\)在圓\(x^2+y^2=4\)上，則\(x+y\)的最大值為（）A.\(2\sqrt{2}\)B.\(4\)C.\(2\)D.\(\sqrt{2}\)答案：A13.直線\(x+y-1=0\)與圓\(x^2+y^2=1\)的位置關(guān)系是（）A.相交且過圓心B.相交不過圓心C.相切D.相離答案：B14.拋物線\(y=4x^2\)上一點(diǎn)\(P\)到焦點(diǎn)的距離為\(1\)，則點(diǎn)\(P\)的縱坐標(biāo)為（）A.\(\frac{15}{16}\)B.\(\frac{31}{32}\)C.\(\frac{63}{64}\)D.\(\frac{127}{128}\)答案：C15.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的左、右焦點(diǎn)分別為\(F1,F2\)，點(diǎn)\(P\)在橢圓上，若\(|PF1|+|PF2|=4\)，\(|F1F2|=2\)，則橢圓的標(biāo)準(zhǔn)方程為（）A.\(\frac{x^2}{4}+\frac{y^2}{3}=1\)B.\(\frac{x^2}{3}+\frac{y^2}{4}=1\)C.\(\frac{x^2}{2}+\frac{y^2}{1}=1\)D.\(\frac{x^2}{1}+\frac{y^2}{2}=1\)答案：A16.雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>b>0)\)的漸近線與圓\(x^2+y^2-6x+5=0\)相切，則雙曲線的離心率為（）A.\(\frac{3\sqrt{5}}{5}\)B.\(\frac{\sqrt{6}}{2}\)C.\(\frac{3}{2}\)D.\(\frac{\sqrt{5}}{2}\)答案：A17.已知直線\(l\)與拋物線\(y^2=8x\)交于\(A,B\)兩點(diǎn)，且\(l\)經(jīng)過拋物線的焦點(diǎn)\(F\)，若\(|AB|=10\)，則直線\(l\)的斜率為（）A.\(\pm1\)B.\(\pm2\)C.\(\pm\frac{1}{2}\)D.\(\pm\frac{1}{4}\)答案：B18.圓\(x^2+y^2-2x-4y+1=0\)關(guān)于直線\(y=x\)對稱的圓的方程為（）A.\((x-2)^2+(y-1)^2=4\)B.\((x+2)^2+(y+1)^2=4\)C.\((x-1)^2+(y-2)^2=4\)D.\((x+1)^2+(y+2)^2=4\)答案：C19.橢圓\(\frac{x^2}{25}+\frac{y^2}{16}=1\)上一點(diǎn)\(P\)到左焦點(diǎn)的距離為\(3\)，則點(diǎn)\(P\)到右準(zhǔn)線的距離為（）A.\(\frac{15}{2}\)B.\(\frac{25}{3}\)C.\(\frac{35}{4}\)D.\(\frac{45}{5}\)答案：B20.雙曲線\(\frac{x^2}{9}-\frac{y^2}{16}=1\)的焦點(diǎn)到漸近線的距離為（）A.2B.3C.4D.5答案：C1.下列方程表示圓的是（）A.\(x^2+y^2=1\)B.\(x^2+y^2-2x+4y=0\)C.\(x^2+y^2+2x-4y+5=0\)D.\(2x^2+2y^2-4x+6y=1\)答案：ABD2.已知直線\(l1:y=2x+1\)，\(l2:y=-2x+1\)，則（）A.\(l1\)與\(l2\)平行B.\(l1\)與\(l2\)垂直C.\(l1\)與\(l2\)的斜率相等D.\(l1\)與\(l2\)的傾斜角互補(bǔ)答案：AD3.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的性質(zhì)有（）A.長軸長為\(2a\)B.短軸長為\(2b\)C.離心率\(e=\frac{c}{a}\)（\(c\)為半焦距）D.焦點(diǎn)坐標(biāo)為\((\pmc,0)\)答案：ABCD4.雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)\)的漸近線方程為（）A.\(y=\pm\frac{a}x\)B.\(y=\pm\frac{a}x\)C.\(ax\pmby=0\)D.\(bx\pmay=0\)答案：AC5.拋物線\(y^2=2px(p>0)\)的焦點(diǎn)坐標(biāo)為（）A.\((\frac{p}{2},0)\)B.\((-\frac{p}{2},0)\)C.\((0,\frac{p}{2})\)D.\((0,-\frac{p}{2})\)答案：A6.已知點(diǎn)\(P(x,y)\)在橢圓\(\frac{x^2}{25}+\frac{y^2}{16}=1\)上，則（）A.\(-4\leqy\leq4\)B.\(-5\leqx\leq5\)C.\(0\leqx\leq5\)D.\(0\leqy\leq4\)答案：AB7.圓\(x^x+y^2=r^2(r>0)\)與直線\(Ax+By+C=0\)相切的條件是（）A.圓心到直線的距離等于半徑B.\(A^2r^2+B^2r^2=C^2\)C.\(\frac{|C|}{\sqrt{A^2+B^2}}=r\)D.\(A^2+B^2=C^2r^2\)答案：ABC8.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)上一點(diǎn)\(P(x0,y0)\)處的切線方程為（）A.\(\frac{x0x}{a^2}+\frac{y0y}{b^2}=1\)B.\(\frac{x0x}{a^2}-\frac{y0y}{b^2}=1\)C.\(b^2x0x+a^2y0y=a^2b^2\)D.\(b^2x0x-a^2y0y=a^2b^2\)答案：AC9.雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)\)的離心率\(e\)的范圍是（）A.\(e>1\)B.\(e\geq1\)C.\(e<\sqrt{2}\)D.\(e\geq\sqrt{2}\)答案：A10.拋物線\(y=ax^2(a\neq0)\)的焦點(diǎn)坐標(biāo)為（）A.\((0,\frac{1}{4a})\)B.\((0,-\frac{1}{4a})\)C.\((\frac{1}{4a},0)\)D.\((-\frac{1}{4a},0)\)答案：A1.平面直角坐標(biāo)系中，兩點(diǎn)間距離公式為\(d=\sqrt{(x2-x1)^2+(y2-y1)^2}\)。（）答案：√2.直線方程的一般式為\(Ax+By+C=0\)（\(A,B\)不同時(shí)為\(0\)）。（）答案：√3.圓的標(biāo)準(zhǔn)方程為\((x-a)^2+(y-b)^2=r^2\)，其圓心坐標(biāo)為\((a,b)\)，半徑為\(r\)。（）答案：√4.橢圓的離心率\(e\)越大，橢圓越圓。（）答案：×5.雙曲線的漸近線方程為\(y=\pm\frac{a}x\)，則雙曲線方程為\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)。（）答案：×6.拋物線\(y^2=2px(p>0)\)的準(zhǔn)線方程為\(x=-\frac{p}{2}\)。（）答案：√7.點(diǎn)\((x0,y0)\)到直線\(Ax+By+C=0\)的距離公式為\(d=\frac{|Ax0+By0+C|}{\sqrt{A^‘+B^2}}\)。（）答案：√8.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)上一點(diǎn)\(P\)到兩焦點(diǎn)距離之和等于\(2a\)。（）答案：√9.雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)\)的實(shí)軸長為\(2b\)。（）答案：×10.拋物線\(y=ax^2(a\neq0)\)的開口