The Planes — Important Questions

x + y + z = a

If intercepts are equal, say a, then the equation is x/a + y/a + z/a = 1, which simplifies to x + y + z = a.

lx + my + nz = p

In the normal form, l, m, n are the direction cosines of the normal to the plane, and p is the length of the perpendicular from the origin.

2, 3, 6

For the plane ax + by + cz + d = 0, the direction ratios of the normal are a, b, c. So here they are 2, 3, 6.

5/13

The length is |d| / √(a² + b² + c²) = |5| / √(3² + 4² + 12²) = 5 / √(9+16+144) = 5/√169 = 5/13.