第八章　平面解析幾何

第八章平面解析幾何[知識網(wǎng)絡(luò)][命題方向]1．解析幾何是高中數(shù)學的重要內(nèi)容．高考主要考查直線與圓、橢圓、拋物線、雙曲線的定義、標準方程和簡單的幾何性質(zhì)．其中直線與圓、直線與圓錐曲線的位置關(guān)系是考查重點．估計在高考中解析幾何試題基本保持為兩道選擇題、一道填空題和一道解答題(或一道選擇題一道填空題和一道解答題)分值為27分(或22分)．2．常見命題方向：(1)圓的方程主要考查定義和性質(zhì)；(2)圓錐曲線(橢圓、拋物線、雙曲線)主要是曲線的定義、標準方程、性質(zhì)(焦點、離心率、準線、漸近線)；(3)以直線與圓錐曲線的位置關(guān)系為背景考查弦長(可以直接用過拋物線焦點的弦長公式，也可以用通用的弦長公式求解)；(4)以直線與圓錐曲線的位置關(guān)系為載體考查定點、定值、取值范圍以及存在性問題；(5)以平面幾何知識或圓錐曲線相關(guān)內(nèi)容為載體考查將平面幾何代數(shù)化的思想．這些內(nèi)容均為高考考查的重點和熱點，因此在備考復習中應加強訓練，同時在備考時還需要加強對數(shù)形結(jié)合思想、分類討論思想、函數(shù)與方程思想在解決圓錐曲線綜合問題中的應用．3．高考對本章的考查方式及題目難度變化不大，延續(xù)此前的考試風格．探究1(人教A版選擇性必修第一冊P52)在平面直角坐標系中，設(shè)直線l的傾斜角為α.(1)已知直線l經(jīng)過O(0，0)，P(eq\r(3)，1)，α與O，P的坐標有什么關(guān)系？(2)類似地，如果直線l經(jīng)過P1(－1，1)，P2(eq\r(2)，0)，α與P1，P2的坐標又有什么關(guān)系？(3)一般地，如果直線l經(jīng)過兩點P1(x1，y1)，P2(x2，y2)，x1≠x2，那么α與P1，P2的坐標有怎樣的關(guān)系？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究2(人教A版選擇性必修第一冊P65)在方程Ax＋By＋C＝0中，A，B，C為何值時，方程表示的直線：①平行于x軸？②平行于y軸？③與x軸重合？④與y軸重合？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究3(人教A版選擇性必修第一冊P74)如圖，已知點P(x0，y0)，直線l：Ax＋By＋C＝0，如何求點P到直線l的距離？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究4(人教A版選擇性必修第一冊P75)我們知道，向量是解決距離、角度問題的有力工具．能否用向量方法求點到直線的距離？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究5(人教A版選擇性必修第一冊P86)方程x2＋y2＋Dx＋Ey＋F＝0中的D，E，F(xiàn)滿足什么條件時，這個方程表示圓？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究6(人教A版選擇性必修第一冊P105)取一條定長的細繩，把它的兩端都固定在圖板的同一點，套上鉛筆，拉緊繩子，移動筆尖，這時筆尖(動點)畫出的軌跡是一個圓．如果把細繩的兩端拉開一段距離，分別固定在圖板的兩點F1，F(xiàn)2(如圖)，套上鉛筆，拉緊繩子，移動筆尖，畫出的軌跡是什么曲線？在這一過程中，移動的筆尖(動點)滿足的幾何條件是什么？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究7(人教A版選擇性必修第一冊P110)觀察橢圓的形狀，可以發(fā)現(xiàn)橢圓既是軸對稱圖形，又是中心對稱圖形，如何利用方程說明橢圓的對稱性？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究8(人教A版選擇性必修第一冊P119)類比求橢圓標準方程的過程，我們?nèi)绾谓⑦m當?shù)淖鴺讼?，得出雙曲線的方程？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究9(人教A版選擇性必修第一冊P121)如圖，點A，B的坐標分別是(－5，0)，(5，0)，直線AM，BM相交于點M，且它們斜率之積是eq\f(4,9)，試求點M的軌跡方程，并由點M的軌跡方程判斷軌跡的形狀，與它們的斜率之積是－eq\f(4,9)比較，你有什么發(fā)現(xiàn)？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題1(人教A版選擇性必修第一冊P58T8)經(jīng)過點P(0，－1)作直線l，若直線l與連接A(1，－2)，B(2，1)兩點的線段總有公共點，求直線l的傾斜角α與斜率k的取值范圍，并說明理由．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題2(人教A版選擇性必修第一冊P63例4)已知△ABC的三個頂點A(－5，0)，B(3，－3)，C(0，2)，求邊BC所在直線的方程，以及這條邊上的中線AM所在直線的方程．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題3(人教A版選擇性必修第一冊P73例4)用坐標法證明：平行四邊形兩條對角線的平方和等于兩條鄰邊的平方和的兩倍．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題4(人教A版選擇性必修第一冊P77例6)已知△ABC的三個頂點分別是A(1，3)，B(3，1)，C(－1，0)，求△ABC的面積．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題5(人教A版選擇性必修第一冊P79T10)已知△ABC的頂點A(5，1)，邊AB上的中線CM所在直線方程為2x－y－5＝0，邊AC上的高BH所在直線方程為x－2y－5＝0.求：(1)頂點C的坐標；(2)直線BC的方程．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題6(人教A版選擇性必修第一冊P89T10)在平面直角坐標系中，如果點P的坐標(x，y)滿足eq\b\lc\{(\a\vs4\al\co1(x＝a＋rcosθ，,y＝b＋rsinθ，))其中θ為參數(shù)．證明：點P的軌跡是圓心為(a，b)，半徑為r的圓．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題7(人教A版選擇性必修第一冊P91例1)已知直線l：3x＋y－6＝0和圓心為C的圓x2＋y2－2y－4＝0，判斷直線l與圓C的位置關(guān)系；如果相交，求直線l被圓C所截得的弦長．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題8(人教A版選擇性必修第一冊P94例4)一個小島的周圍有環(huán)島暗礁，暗礁分布在以小島中心為圓心，半徑為20km的圓形區(qū)域內(nèi)．已知小島中心位于輪船正西40km處，港口位于小島中心正北30km處．如果輪船沿直線返港，那么它是否會有觸礁危險？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題9(人教A版選擇性必修第一冊P103T20)已知圓C：(x－1)2＋(y－2)2＝25，直線l：(2m＋1)x＋(m＋1)y－7m－4＝0.(1)求證：直線l恒過定點．(2)直線l被圓C截得的弦何時最長、何時最短？并求截得的弦長最短時m的值以及最短弦長．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題10(人教A版選擇性必修第一冊P108例2)如圖，在圓x2＋y2＝4上任取一點P，過點P作x軸的垂線段PD，D為垂足．當點P在圓上運動時，線段PD的中點M的軌跡是什么？為什么？________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題11(人教A版選擇性必修第一冊P128T12)設(shè)橢圓eq\f(x2,a2)＋eq\f(y2,b2)＝1(a＞b＞0)與雙曲線eq\f(x2,a2)－eq\f(y2,b2)＝1的離心率分別為e1，e2，雙曲線的漸近線的斜率小于eq\f(2\r(5),5)，求e1和e2的取值范圍．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·全國甲卷)記雙曲線C：eq\f(x2,a2)－eq\f(y2,b2)＝1(a＞0，b＞0)的離心率為e，寫出滿足條件“直線y＝2x與C無公共點”的e的一個值________．點評本題與教材習題都是考查的雙曲線的離心率的取值范圍．典題12(人教A版選擇性必修第一冊P128T13)已知雙曲線x2－eq\f(y2,2)＝1，過點P(1，1)的直線l與雙曲線相交于A，B兩點，P能否是線段AB的中點？為什么？___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題13(人教A版選擇性必修第一冊P135例4)斜率為1的直線l經(jīng)過拋物線y2＝4x的焦點F，且與拋物線相交于A，B兩點，求線段AB的長．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題14(人教A版選擇性必修第一冊P136例5)經(jīng)過拋物線焦點F的直線交拋物線于A，B兩點，經(jīng)過點A和拋物線頂點的直線交拋物線的準線于點D，求證：直線DB平行于拋物線的對稱軸．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題15(人教A版選擇性必修第一冊P146T10)如圖，已知直線與拋物線y2＝2px(p＞0)交于A，B兩點，且OA⊥OB，OD⊥AB交于點D，點D的坐標為(2，1)，求p的值．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題16(人教A版選擇性必修第一冊P103T19)一條光線從點A(－2，3)射出，經(jīng)x軸反射后，與圓C：(x－3)2＋(y－2)2＝1相切，求反射后光線所在直線的方程．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2015·山東卷)一條光線從點(－2，－3)射出，經(jīng)y軸反射后與圓(x＋3)2＋(y－2)2＝1相切，則反射光線所在直線的斜率為()A．－eq\f(5,3)或－eq\f(3,5) B．－eq\f(3,2)或－eq\f(2,3)C．－eq\f(5,4)或－eq\f(4,5) D．－eq\f(4,3)或－eq\f(3,4)點評本題考查了反射光線的性質(zhì)，直線與圓相切的性質(zhì)、點到直線的距離公式、點斜式、對稱點，與課本習題命題角度類似，屬于改編題．典題17(人教A版選擇性必修第一冊P108例3)如圖，設(shè)A，B兩點的坐標分別為(－5，0)，(5，0)．直線AM，BM相交于點M，且它們的斜率之積是－eq\f(4,9)，求點M的軌跡方程．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2019·全國Ⅱ卷節(jié)選)已知點A(－2，0)，B(2，0)，動點M(x，y)滿足直線AM與BM的斜率之積為－eq\f(1,2).記M的軌跡為曲線C.求C的方程，并說明C是什么曲線．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________點評本題利用斜率公式表示斜率，然后化簡成橢圓的標準方程，與課本習題命題角度類似，屬于改編題．典題18(人教A版選擇性必修第一冊P114T2)經(jīng)過橢圓eq\f(x2,2)＋y2＝1的左焦點F1作傾斜角為60°的直線l，直線l與橢圓相交于A，B兩點，求線段AB的長．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2010·遼寧卷)設(shè)橢圓C：eq\f(x2,a2)＋eq\f(y2,b2)＝1(a＞b＞0)的左焦點為F，過點F的直線l與橢圓C相交于A，B兩點，直線l的傾斜角為60°，eq\o(AF,\s\up6(→))＝2eq\o(FB,\s\up6(→)).(1)求橢圓C的離心率；(2)如果|AB|＝eq\f(15,4)，求橢圓C的方程．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________點評本題考查橢圓的性質(zhì)和標準方程，以及直線和圓錐曲線的位置關(guān)系，與課本習題命題角度類似，屬于改編題．典題19(人教A版選擇性必修第一冊P126例6)如圖，過雙曲線eq\f(x2,3)－eq\f(y2,6)＝1的右焦點F2，傾斜角為30°的直線交雙曲線于A，B兩點，求|AB|._____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________