Question

If a right circular cone has three mutually perpendicular generators,prove that the semi-vertical angle is tan^-1(√2)​

Answer 1

Given:  Right circular cone has three mutually perpendicular generators

To find: Prove that the semi-vertical angle is tan^-1(√2)​.

Solution:

• Now we have given that a right circular cone has three mutually perpendicular generators.

• So, ax² + by² + cz² + 2fyx + 2gzx + 2hxy = 0, this means it has a+b+c=0.

• Now, lets assume three mutually perpendicular generators with direction cosines as l(i), m(i), n(i), where i = 1, 2, 3.

• Then the direction cosines will be:

               ∑l1/3 , ∑m1/3 , ∑n1/3 = L, M, N

• Now since the three generators are mutually perpendicular to each other, so we get:

               l(i)l(j) + m(i)m(j) + n(i)n(j) = 0   .......where i ≠ j

• So from the above terms, we get:

               l1m1 + l2m2 + l3m3 = 0

• We know that

               cos x = l1 x L + m1 x M + n1 x N / (√L²+M²+N²)  =  1/√3

• By solving this, we get:

               x = tan -1 (√2)

Answer:

                  So from above, we have proved that the semi-vertical angle is tan^-1(√2)​.