If in a right angled triangle, `a\ a n d\ b` are the lengths of sides and `c` is the length of hypotenuse and `c-b!=1,\ c+b!=1` , then show that `(log)_("c"+"b")"a"+(log)_("c"-"b")=2(log)_("c"+"b")adot(log)_("c"-"b")adot`

Options

A. 1
B. 2
C. `(1)/(2)`
D. none of these

Answer

B

Explanation

Since a, b, c are the sides of a right-angled triangle with c as the largest side i.e. hypotenuse. Therefore,
`c^(2) = a^(2) + b^(2)`
Now,
`("log"_(c+b)a + "log"_(c-b)a)/("log"_(c+b)a."log"_(c-b)a)`
`= (1)/("log"_((c-b))a) + (1)/("log"_((c+b))a) = "log"_(a) (c-b) + "log"_(a) (c+b)`
` = "log"_(a) (c^(2) -b^(2)) = "log"_(a)a^(2) = 2"log"_(a)a = 2`