If 1a + 1b = k and ab = 1m, find 1a^2 + 1b^2 | GMAT Sprint

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555-605

Algebra

If $\frac{1}{a} + \frac{1}{b} = k$ and $ab = \frac{1}{m}$ , find $\frac{1}{a^2} + \frac{1}{b^2}$

Answer Choices

Correct answer marked below

\frac{1}{k^2} - \frac{2}{m}

k^2 - \frac{2}{m}

k^2 - 2m

Correct

\frac{1}{k^2} - 2m

\frac{1}{k^2} + 2m

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Explanation

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Official Explanation

Official

To find

\frac{1}{a^2} + \frac{1}{b^2}

, we can use the identity

\frac{1}{a^2} + \frac{1}{b^2} = \left(\frac{1}{a} + \frac{1}{b}\right)^2 - 2\left(\frac{1}{a}\cdot\frac{1}{b}\right)

. Given that

\frac{1}{a} + \frac{1}{b} = k

, we have [m]\left(\frac{1}{a} +…

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Comments

ChengchenApr 13, 2025

Can you explain more?

DevanshJan 23, 2025

Great question!

NguyenApr 20, 2025

greatt question!

lucianMar 4, 2025

Amazing

chungMar 1, 2025

Great

What this question tests

What it tests

Your ability to manipulate linear and quadratic expressions, substitute unknowns, and solve equations under time pressure.

Common trap

Losing a sign when moving terms across the equals sign, or dividing both sides by a variable that could equal zero.

If 1a+1b=k\frac{1}{a} + \frac{1}{b} = ka1​+b1​=k and ab=1mab = \frac{1}{m}ab=m1​, find 1a2+1b2\frac{1}{a^2} + \frac{1}{b^2}a21​+b21​