Prove that the st. line ax + by + c = 0 divides the join of `(x_1, y_1)` and `(x_2, y_2)` in the ratio `-(ax_1 + by_1 +c)/(ax_2+ by_2+c)`.

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))

Find the ratio in which the plane ax+by+cz+d=0 divides the join of the points P(x_1,y_1,z_1) and Q(x_2,y_2,z_2) lying on the plane. Hence, show that the points P(1,-2,3) and Q(0,0,-1) lie on opposite sides of the plane 2x+5y+7z=3.

Prove that st. line 5x -2y - 1 = 0 is mid-parailel to the st. lines : 5x - 2y -9 =0 and 5x - 2y + 7 = 0.

Prove that the equation of the st. line perpendicular to Ax + By + C = 0 and passing through (x_1,y_1) is Bx- Ay = Bx_1 - Ay_1 .

Prove that the equation of the st. line parallel to Ax + By + C = 0 and passing through (x_1,y_1) is A(x-x_1)+B(y-y_1)=0 .

Prove that the line through the point (x_1,y_1) and parallel to the line Ax + By + C = 0 is A (x- x_1) + B (y -y_1) = 0.

Write the following equation in the form ax + by + c = 0 and indicate the values of a, b and c in the case : 2x=y .

Write the following equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case: 2x = y

Prove that the line through the point (x_(1),y_(1)) and parallel to the line Ax+By+C=0 is A(x-x_(1)) +B(y-y_(1))=0 .

Statement 1 :If the point (2a-5,a^2) is on the same side of the line x+y-3=0 as that of the origin, then a in (2,4) Statement 2 : The points (x_1, y_1)a n d(x_2, y_2) lie on the same or opposite sides of the line a x+b y+c=0, as a x_1+b y_1+c and a x_2+b y_2+c have the same or opposite signs.