Ans. 0005

Sol. The required line through the point P(2, 1, 1) in the plane 2x + y –5z = 0 and of greatest slope is perpendicular to line of intersection of the planes

2x + y – 5z = 0 and 4x – 3y + 7z = 0

Let dr's. of line of intersection are a, b, c then

2a + b –5c = 0 and 4a –3b + 7c = 0

as dr's of line (a, b, c) is perpendicular to dr's. of

normal to both the planes.

= =

Now let the dr's of required line be λ , m, n

then = = where 2 λ + m – 5n = 0

4 λ + 17m + 5n = 0

So = = ⇒ = = = k

Let any point on this line ≡ (2 + 3k, 1 – k, 1 + k)

also on first plane

4 (2 + 3k) – 3(1 – k) + 7 (1 + k) = 0

⇒ 8 + 12k – 3 + 3k + 7 + 7k = 0

⇒ k =

so point ≡

So x + y + z = 4 +

greatest integral value = 5