The orthocenter of triangle whose vertices are `A(a,0,0),B(0,b,0)` and `C(0,0,c)` is `(k/a,k/b,k/c)` then k is equal to

Options

A. `(1/a^(2)+1/b^(2)+1/c^(2))^(-1)`
B. `(1/a+1/b+1/c)^(-1)`
C. `(1/a^(2)+1/b^(2)+1/c^(2))`
D. `(1/a+1/b+1/c)`

Answer

A

Explanation

Let `O(alpha,beta,gamma)` be orthocenter.
Now, `AO bot BC`
Similarly, `a(alpha)=b(beta)=cgamma=k`
`therefore alpha=k/a, beta=k/b,gamma=k/c`
The plane `x/a+y/b+z/c=1` contains `(alpha,beta,gamma)`
`therefore k(1/a^(2)+1/b^(2)+1/c^(2))=1`