Cho 4 điểm ABCD Viết phương trình mặt phẳng đi qua A và song song với mặt phẳng bcd

Trong không gian Oxyz , cho bốn điểm A0;1;−1 , B1;1;2 , C1;−1;0 và D0;0;1 . Mặt phẳng α song song với mặt phẳng BCD và chia khối tứ diện ABCD thành hai khối đa diện sao cho tỉ số thể tích của khối đa diện có chứa điểm A và khối tứ diện ABCD bằng 127 . Viết phương trình mặt phẳng α .

A.y+z−4=0 .

B.y−z−1=0 .

C.−y+z−4=0 .

D.3x−3z−4=0 .

Đáp án và lời giải

Đáp án:B

Lời giải:Li gii
Chn B

Cho 4 điểm ABCD Viết phương trình mặt phẳng đi qua A và song song với mặt phẳng bcd

Gọi M , N , P lần lượt là giao điểm của mặt phẳng α với các cạnh AB , AC , AD . Đặt AMAB=k , 0 Ta có: VAMNPVABCD=AMAB. ANAC. APAD=k3=127 ⇒k=13 .
Ta có: AM→=13. AB→ ⇒M13;1;0 .
BC→=0;−2;−2, BD→=−1;−1;−1 ⇒n→=12. BC→,BD→=0;1;−1 .
Mặt phẳng α đi qua M và song song với mặt phẳng BCD nên nhận véc-tơ n→=0;1;−1 làm véc-tơ pháp tuyến có phương trình là: 0x−13+1y−1−1z−0=0
hay y−z−1=0 .

Vậy đáp án đúng là B.

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