Show that the plane `ax+by+cz+d=0` divides the line joining `(x_1, y_1, z_1) and (x_2, y_2, z_2)` in the ratio of `(-(ax_1+ay_1+cz_1+d)/(ax_2+by_2+cz_2+d))`

Show that the plane a x+b y+c z+d=0 divides the line joining the points (x_1, y_1, z_1)a n d(x_2, y_2, z_2) in the ratio (a x_1+b y_1+c z_1+d)/(a x_2+b y_2+c z_2+d) .

The ratio in which the line segment joining P(x_1,\ y_1) and Q(x_2,\ y_2) is divided by x-axis is (a) y_1: y_2 (b) y_1: y_2 (c) x_1: x_2 (d) x_1: x_2

Show that the coordinates off the centroid of the triangle with vertices A(x_1, y_1, z_1),\ B(x_2, y_2, z_2)a n d\ (x_3, y_3, z_3) are ((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3,(z_1+z_2+z_3)/3)

The ratio in which the plane y-1=0 divides the straight line joining (1,-1,3) and (-2,5,4) is (A) 1:2 (B) 3:1 (C) 5:2 (D) 1:3

Show that the equation of the chord of the parabola y^2 = 4ax through the points (x_1, y_1) and (x_2, y_2) on it is : (y-y_1) (y-y_2) = y^2 - 4ax

Show that the area of the triangle having vertices (0,0,0),(x_1,y_1,z_1),(x_2,y_2,z_2) is 1/2 sqrt((y_1z_2-y_2z_1)^2+(z_2x_1-x_2z_1)^2+(x_1y_2-x_2y_1)^2) .

Find the angle between the line joining the points (3,\ -4,\ -2)a n d\ (12 ,\ 2,\ 0) and the plane 3x-y+z=1.

If the plane ax – by + cz = d contains the line (x – a)/a = (y – 2d)/b = (z – c)/c, then the value of (b/4d) is equal to (b, d ne 0)

Show that if the axes are rectangular the equation of line through point (x_1,y_1,z_1) at right angle to the lines x/l_1=y/m_1=z/n_1,x/l_2=y/m_2=z/n_2 is (x-x_1)/(m_1n_2-m_2n_1)=(y-y_1)/(n_1l_2-n_2l_1)=(z-z_1)/(l_1m_2-l_2m_1)

Find the reflection of the plane ax+by+cz+d=0 in the plane a\'x+b\'y+c\'z+d\'=0