Question

If \( (a^2 + b^2), (b^2 + c^2) \) and \( (a^2 + c^2) \) are in geometric progression, which of the following holds true?

• A. \( b^2 - c^2 = \dfrac{a^2 - c^2}{b^2 + a^2} \)
• B. \( b^2 - a^2 = \dfrac{a^2 - c^2}{b^2 + c^2} \)
• C. \( b^2 - c^2 = \dfrac{a^2 - c^2}{b^2 + a^2} \)
• D. \( a^2 - b^2 = \dfrac{b^2 + c^2}{b^2 + a^2} \)

Hint:

Use \( y^2 = xz \) when three terms are in geometric progression and plug values directly.

Answer 1

The Correct Option is C

If \( x, y, z \) are in GP, then \( y^2 = xz \).
Let: \[ x = a^2 + b^2,\quad y = b^2 + c^2,\quad z = a^2 + c^2 \] Then apply the identity: \[ (b^2 + c^2)^2 = (a^2 + b^2)(a^2 + c^2) \] Expanding and simplifying leads to the identity: \[ b^2 - c^2 = \frac{a^2 - c^2}{b^2 + a^2} \] So the correct relation is: \[ \boxed{b^2 - c^2 = \dfrac{a^2 - c^2}{b^2 + a^2}} \]