Cho (P) : (4 -$ \alpha$ )x - ($ \alpha$ +5)y +$ \alpha$ z + $\alpha$ =0 . (Q) : 2x + 3y + mz +5 =0 . (R) : 3x+ny+ $\alpha ( n- \alpha )$z + n = 0
a) tìm m, $\alpha $ để (P) song song (Q)
b) tìm n, $\alpha $ để (P) song song (R)
c) tìm mối liên hệ giữa $ \alpha $ , m để ( P) vuông góc (Q)
c.
Vec-tơ pháp tuyến của $(P)$ là: $\overrightarrow {n_1}=(4-\alpha;-5-\alpha;\alpha)$
Vec-tơ pháp tuyến của $(Q)$ là: $\overrightarrow {n_2}=(2;3;m)$
Ta có: $(P)\perp(Q)$
$\Leftrightarrow \overrightarrow {n_1}.\overrightarrow {n_2}=0$
$\Leftrightarrow 2(4-\alpha)+3(5-\alpha)+m\alpha=0$
$\Leftrightarrow m\alpha-5\alpha+23=0$
b.
Vec-tơ pháp tuyến của $(P)$ là: $\overrightarrow {n_1}=(4-\alpha;-5-\alpha;\alpha)$
Vec-tơ pháp tuyến của $(R)$ là: $\overrightarrow {n_3}=(3;n;\alpha(n-\alpha))$
Ta có: $(P)//(R)$
$\Leftrightarrow \overrightarrow {n_1},\overrightarrow {n_3}$ cùng phương
$\Leftrightarrow \frac{4-\alpha}{3}=\frac{-5-\alpha}{n}=\frac{\alpha}{\alpha(n-\alpha)}$
$\Leftrightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l} \alpha=-2\\n=-\frac{3}{2} \end{array} \right.\\\left\{ \begin{array}{l} \alpha=9\\n=\frac{42}{5} \end{array} \right. \end{array} \right.$
a.
Vec-tơ pháp tuyến của $(P)$ là: $\overrightarrow {n_1}=(4-\alpha;-5-\alpha;\alpha)$
Vec-tơ pháp tuyến của $(Q)$ là: $\overrightarrow {n_2}=(2;3;m)$
Ta có: $(P)//(Q)$
$\Leftrightarrow \overrightarrow {n_1},\overrightarrow {n_2}$ cùng phương
$\Leftrightarrow \frac{4-\alpha}{2}=\frac{-5-\alpha}{3}=\frac{\alpha}{m}$
$\Leftrightarrow \left\{ \begin{array}{l} \alpha=22\\m=-\frac{22}{9} \end{array} \right.$

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