Question

If \(x\), y, z are the sides of a right angled triangle, where z is the largest side, then \(\frac{1}{log_{x+z}y}+\frac{1}{log_{z-x}y} =\)

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(4\)

Hint:

Use 1/log_a b = log_b a to combine the two terms, then Pythagoras.

Answer 1

The Correct Option is B

Step 1: Flip the logs
$1/\log_a y = \log_y a$, so the sum becomes $\log_y(x+z) + \log_y(z-x)$.

Step 2: Multiply inside
This equals $\log_y(z^2 - x^2)$.

Step 3: Apply Pythagoras
As $z^2 = x^2 + y^2$, we get $z^2 - x^2 = y^2$ and $\log_y y^2 = 2$. Option (B).

Final Answer:
Option (B). \[ \boxed{2} \]