Question

Numbers 𝑎,𝑏 and 𝑐 are in continued proportion.

Statement (1): (𝑎 + 𝑏 + 𝑐)⁢(𝑎 − 𝑏 + 𝑐) = \[𝑎^2+𝑏^2+𝑐^2\]

Statement (2): \[𝑏^2 = 𝑎⁢𝑐\] and (𝑎 +𝑏 + 𝑐) ⁢(𝑎 − 𝑏 + 𝑐) = \[(𝑎+𝑐)^2 −𝑏^2\]

Options

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Both the statements are true.

Explanation:

Numbers a, b, c are in continued proportion.

Now, (a + b + c)(a − b + c)

= [(a + c) + b][(a + c) − b]

= (a + c)² − b²

= a² + c² + 2ac − b²

Given, a, b and c are in continued proportion,

∴ `a/b = b/c`

⇒ b² = ac

Substituting the value b² = ac in above equation,

= a² + c² + 2b² − b²

= a² + c² + b²

So, Statement 1 correctly states that: (a + b + c)(a - b + c) = a² + b² + c².

and

Statement 2 correctly states that: b² = ac