橢圓離心率試題及答案

一、單項選擇題(每題2分,共10題)1.橢圓\(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\)的離心率是(）A.\(\frac{\sqrt{5}}{3}\)B.\(\frac{\sqrt{5}}{2}\)C.\(\frac{2}{3}\)D.\(\frac{3}{2}\)2.已知橢圓\(C\):\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的半焦距為\(c\),離心率\(e=\frac{1}{2}\),則\(\frac{a}\)的值為()A.\(\frac{\sqrt{2}}{2}\)B.\(\frac{\sqrt{3}}{3}\)C.\(\frac{\sqrt{3}}{2}\)D.\(\frac{1}{2}\)3.橢圓\(25x^{2}+16y^{2}=400\)的離心率為()A.\(\frac{3}{5}\)B.\(\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(\frac{16}{25}\)4.若橢圓的一個焦點與短軸的兩個端點構(gòu)成一個正三角形,則該橢圓的離心率為()A.\(\frac{1}{2}\)B.\(\frac{\sqrt{3}}{2}\)C.\(\frac{\sqrt{3}}{3}\)D.\(\frac{\sqrt{6}}{3}\)5.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的左、右焦點分別為\(F_1,F_2\),若橢圓上存在一點\(P\)使得\(\angleF_1PF_2=90^{\circ}\),且\(|PF_1|=2|PF_2|\),則橢圓的離心率為()A.\(\frac{\sqrt{5}}{3}\)B.\(\frac{2}{3}\)C.\(\frac{\sqrt{2}}{3}\)D.\(\frac{1}{3}\)6.橢圓\(\frac{x^{2}}{m}+\frac{y^{2}}{4}=1\)的離心率為\(\frac{1}{2}\),則\(m\)的值為()A.\(\frac{16}{3}\)B.\(\frac{3}{16}\)C.\(\frac{16}{3}\)或\(3\)D.\(\frac{3}{16}\)或\(3\)7.已知橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率\(e=\frac{\sqrt{3}}{2}\),且過點\((2,1)\),則橢圓方程為()A.\(\frac{x^{2}}{8}+\frac{y^{2}}{2}=1\)B.\(\frac{x^{2}}{4}+y^{2}=1\)C.\(\frac{x^{2}}{12}+\frac{y^{2}}{3}=1\)D.\(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\)8.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率\(e=\frac{1}{2}\),左焦點為\(F\),\(A,B\)分別為橢圓的上、下頂點,\(M\)是橢圓上一點且不與\(A,B\)重合,則直線\(MA\)與直線\(MB\)的斜率之積為()A.\(-\frac{3}{4}\)B.\(-\frac{4}{3}\)C.\(\frac{3}{4}\)D.\(\frac{4}{3}\)9.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率為\(\frac{\sqrt{2}}{2}\),則雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)的離心率為()A.\(\sqrt{2}\)B.\(\sqrt{3}\)C.\(\frac{\sqrt{6}}{2}\)D.\(\frac{\sqrt{5}}{2}\)10.已知橢圓\(C\):\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的左、右焦點分別為\(F_1,F_2\),離心率為\(\frac{1}{2}\),若以\(F_1F_2\)為直徑的圓與橢圓\(C\)有交點,則橢圓\(C\)的離心率的取值范圍是()A.\((0,\frac{1}{2}]\)B.\([\frac{1}{2},1)\)C.\((0,\frac{\sqrt{2}}{2}]\)D.\([\frac{\sqrt{2}}{2},1)\)答案:1.A2.C3.A4.A5.A6.C7.A8.A9.C10.B二、多項選擇題(每題2分,共10題)1.下列關(guān)于橢圓離心率的說法正確的是(）A.橢圓離心率的范圍是\((0,1)\)B.橢圓離心率越大,橢圓越扁C.橢圓離心率越小,橢圓越圓D.橢圓離心率\(e=\frac{c}{a}\)(\(c\)為半焦距,\(a\)為長半軸長)2.已知橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率為\(\frac{1}{3}\),則（）A.\(a=3b\)B.\(a^{2}=\frac{9}{8}b^{2}\)C.\(b^{2}=\frac{8}{9}a^{2}\)D.\(c=\frac{1}{3}a\)(\(c\)為半焦距)3.橢圓\(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\)的相關(guān)性質(zhì)正確的是(）A.長軸長為\(4\)B.短軸長為\(3\)C.離心率為\(\frac{1}{2}\)D.焦點坐標(biāo)為\((\pm1,0)\)4.若橢圓\(\frac{x^{2}}{m}+\frac{y^{2}}{n}=1(m>0,n>0)\)的離心率為\(\frac{1}{2}\),則()A.當(dāng)焦點在\(x\)軸上時,\(m=\frac{4}{3}n\)B.當(dāng)焦點在\(x\)軸上時,\(n=\frac{3}{4}m\)C.當(dāng)焦點在\(y\)軸上時,\(m=\frac{3}{4}n\)D.當(dāng)焦點在\(y\)軸上時,\(n=\frac{4}{3}m\)5.橢圓\(C\)：\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\),若滿足（）,則離心率\(e=\frac{\sqrt{2}}{2}\)A.\(a^{2}=2b^{2}\)B.\(b^{2}=2a^{2}\)C.\(a=\sqrt{2}b\)D.\(c=b\)(\(c\)為半焦距)6.已知橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率為\(e\),左、右焦點分別為\(F_1,F_2\),\(P\)是橢圓上一點,下列說法正確的是（）A.\(|PF_1|+|PF_2|=2a\)B.\(|PF_1|\cdot|PF_2|\leqa^{2}\)C.若\(\trianglePF_1F_2\)為直角三角形,則離心率\(e\)的取值范圍與直角頂點位置有關(guān)D.離心率\(e\)越大,\(|PF_1|\)與\(|PF_2|\)的和越小7.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率\(e\)滿足\(\frac{1}{3}<e<\frac{1}{2}\),則（）A.\(a^{2}\)與\(b^{2}\)的關(guān)系不確定B.\(a^{2}>2b^{2}\)C.\(a^{2}<\frac{9}{8}b^{2}\)D.\(b^{2}<\frac{8}{9}a^{2}\)8.對于橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)和橢圓\(\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1(A>B>0)\),若它們離心率相同,則()A.\(\frac{a}=\frac{B}{A}\)B.\(\frac{b^{2}}{a^{2}}=\frac{B^{2}}{A^{2}}\)C.\(a^{2}-b^{2}=A^{2}-B^{2}\)D.兩橢圓形狀相似9.已知橢圓\(C\)的離心率為\(e\),下列說法正確的是()A.若\(e=\frac{1}{2}\),則橢圓\(C\)可能是\(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\)B.若\(e=\frac{\sqrt{2}}{2}\),則橢圓\(C\)可能是\(\frac{x^{2}}{2}+y^{2}=1\)C.若\(e=\frac{\sqrt{3}}{2}\),則橢圓\(C\)可能是\(\frac{x^{2}}{4}+y^{2}=1\)D.離心率\(e\)決定了橢圓的形狀10.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率\(e\)與長軸長\(2a\)、短軸長\(2b\)的關(guān)系,下列說法正確的是()A.\(e^{2}=1-\frac{b^{2}}{a^{2}}\)B.\(e\)越大,\(a\)與\(b\)的差距越大C.\(e\)越小,\(2a\)與\(2b\)越接近D.\(e\)與\(a,b\)沒有直接關(guān)系答案:1.ABCD2.CD3.ACD4.BC5.ACD6.ABC7.BD8.ABD9.ABD10.ABC三、判斷題(每題2分,共10題)1.橢圓\(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\)的離心率為\(\frac{\sqrt{7}}{4}\)。()2.橢圓離心率\(e\)越大,橢圓越接近圓。(）3.若橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率為\(\frac{1}{2}\),則\(a=2b\)。（)4.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)與\(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1(a>b>0)\)離心率相同。()5.橢圓離心率的大小只與橢圓的形狀有關(guān),與橢圓的位置無關(guān)。()6.已知橢圓的離心率為\(0.8\),則該橢圓是比較扁的。()7.若橢圓\(C\)的離心率\(e=0\),則橢圓\(C\)為圓。(）8.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)中,離心率\(e\)滿足\(e=\frac{c}{a}\),\(c^{2}=a^{2}-b^{2}\)。（)9.離心率\(e=\frac{\sqrt{3}}{3}\)的橢圓比離心率\(e=\frac{\sqrt{2}}{2}\)的橢圓更圓。()10.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)上一點\(P\)到兩焦點距離之和為\(2a\),離心率與這個距離和無關(guān)。（）答案:1.√2.×3.×4.√5.√6.√7.√8.√9.√10.√四、簡答題(每題5分,共4題)1.求橢圓\(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)的離心率。答案:在橢圓\(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)中,\(a^{2}=25\),\(a=5\);\(b^{2}=16