504 Sia \( A B C D \) un rettangolo in cui \( \overline{A B}=6 a \) e \( \overline{B C}=2 a \). Siano \( M \) ed \( N \), rispettivamente, i punti medi di \( A D \) e \( B C \). Determina, sul segmento \( M N \), due punti \( P \) e \( Q \), con \( M P<M Q \), in modo che: a. l'area del trapezio \( P Q C D \) superi di \( 5 a^{2} \) l'area del triangolo \( A P D \); b. l'area del triangolo \( B C Q \) sia \( \frac{1}{12} \) dell'area del rettan- golo \( A B C D \).

What results when a plane is parallel to the axis of a double-napped cone and passes through the vertex? point parabola two intersecting lines a single line

Find the perpendicular d \( x^{2}+y^{2}-8 x+2 y+8 \) Hence, find the shortest

Determine if the following lines are parallel, perpendicular, or neither \[ \begin{array}{l}y=\frac{8 x}{7}-1 \\ -x+3 y=1\end{array} \]

Find the equation of line perpendicular to \( 5 x-3 y+4=0 \) that passes through the point \( (-4,1) \).

BCD un carré et \( G= \) bary \( \{(A, 2) ;(B,-1) ;(C, 2) ;(D ; 1)\} \) oient \( I= \) bary \( \{(A, 2) ;(B,-1)\} \) et \( J= \) bary \( \{(C, 2) ;(D ; 1)\} \); Montrer que \( 2 \overrightarrow{G A}-\overrightarrow{G B}=\overrightarrow{G I} \) et \( 2 \overrightarrow{G C}+\overrightarrow{G D}=3 \overrightarrow{G j} \) En déduire que \( G=\operatorname{bary\{ }(I, 1) ;(1,3)\} \) Construire I et \( J \) puis \( G \) a) Vérifier que: \( \forall M \in(P): 2 \overrightarrow{M A}-\overrightarrow{M B}+2 \overrightarrow{M C}+\overrightarrow{M D}=4 \overrightarrow{M G} \) b) Déterminer l'ensemble des points M du plan \( (P) \) tel que : \[ \|2 \overrightarrow{M A}-\overrightarrow{M B}+2 \overrightarrow{M C}+\overrightarrow{M D}\|=4\|2 \overrightarrow{M A}-\overrightarrow{M B}\| \]