極坐標(biāo)系中橢圓C的方程為

以極點(diǎn)為原點(diǎn),極軸為

軸非負(fù)半軸,建立平面直角坐標(biāo)系,且兩坐標(biāo)系取相同的單位長(zhǎng)度.
(Ⅰ)求該橢圓的直角標(biāo)方程;若橢圓上任一點(diǎn)坐標(biāo)為

,求

的取值范圍;
(Ⅱ)若橢圓的兩條弦

交于點(diǎn)

,且直線

與

的傾斜角互補(bǔ),
求證:

.
(Ⅰ)

(Ⅱ)詳見(jiàn)解析
試題分析:將橢圓的極坐標(biāo)方程轉(zhuǎn)化為一般標(biāo)準(zhǔn)方程,再利用換元法求范圍,利用參數(shù)方程代入,計(jì)算得到結(jié)果.
試題解析:(Ⅰ)該橢圓的直角標(biāo)方程為

, 2分
設(shè)

,

所以

的取值范圍是

4分
(Ⅱ)設(shè)直線

的傾斜角為

,直線

的傾斜角為

,

則直線

的參數(shù)方程為

(

為參數(shù)),(5分)
代入

得:

即

7分
同理

9分
所以

(10分)
練習(xí)冊(cè)系列答案
相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
曲線

在矩陣

的變換作用下得到曲線

.
(Ⅰ)求矩陣

;
(Ⅱ)求矩陣

的特征值及對(duì)應(yīng)的一個(gè)特征向量.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
已知橢圓

的離心率為

,橢圓的短軸端點(diǎn)與雙曲線

的焦點(diǎn)重合,過(guò)點(diǎn)

且不垂直于

軸直線

與橢圓

相交于

、

兩點(diǎn).
(Ⅰ)求橢圓

的方程;
(Ⅱ)求

的取值范圍.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
已知點(diǎn)

的坐標(biāo)分別是

、

,直線

相交于點(diǎn)

,且它們的斜率之積為

.
(1)求點(diǎn)

軌跡

的方程;
(2)若過(guò)點(diǎn)

的直線

與(1)中的軌跡

交于不同的兩點(diǎn)

,試求

面積的取值范圍(

為坐標(biāo)原點(diǎn)).
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
如圖,曲線

與曲線

相交于

、

、

、

四個(gè)點(diǎn).
⑴ 求

的取值范圍;
⑵ 求四邊形

的面積的最大值及此時(shí)對(duì)角線

與

的交點(diǎn)坐標(biāo).

查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:單選題
已知雙曲線

的左、右焦點(diǎn)分別為

,以

為直徑的圓與雙曲線漸近線的一個(gè)交點(diǎn)為

,則此雙曲線的方程為( )
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:填空題
設(shè)拋物線

上一點(diǎn)

到

軸的距離是

,則點(diǎn)

到該拋物線焦點(diǎn)的距離是____.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:填空題
下列關(guān)于圓錐曲線的命題:其中真命題的序號(hào)___________.(寫出所有真命題的序號(hào))。
① 設(shè)

為兩個(gè)定點(diǎn),若

,則動(dòng)點(diǎn)

的軌跡為雙曲線;
② 設(shè)

為兩個(gè)定點(diǎn),若動(dòng)點(diǎn)

滿足

,且

,則

的最大值為8;
③ 方程

的兩根可分別作橢圓和雙曲線的離心率;
④ 雙曲線

與橢圓

有相同的焦點(diǎn)
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
已知橢圓

的中心在原點(diǎn),焦點(diǎn)在

軸上.若橢圓上的點(diǎn)

到焦點(diǎn)

、

的距離之和等于4.
(1)寫出橢圓

的方程和焦點(diǎn)坐標(biāo).
(2)過(guò)點(diǎn)

的直線與橢圓交于兩點(diǎn)

、

,當(dāng)

的面積取得最大值時(shí),求直線

的方程.
查看答案和解析>>