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

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PS · Problem Solving

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

KyathariSep 28

classic symmetric identity, just square the first equation and swap in ab. took me 20 sec but i nearly picked B out of habit lol

EnaOct 12

same here, i always forget to divide by the product. the 2/m vs 2m trap got me the first time

BayazidMay 4

wait why does the 2/ab term turn into 2m and not 2/m? i keep messing this up

AlessandroMay 18

because they gave you ab = 1/m, so 1/ab = m. flip it

ShivangMay 28

a and b could be negative too right? does that change anything here

SugamMay 31

nah the identity holds regardless of sign, squaring kills the sign issue

AzharAug 2

got C but spent way too long second-guessing between C and D. decent question for the 600 range

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​