If $ a + b = 12 $ and $ ab = 8 $ then the numerical value of $... | Mathematics MCQ | Examoogle

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If $a + b = 12$ and $ab = 8$ then the numerical value of $\left( \frac{a}{b} + \frac{b}{a} \right)$ will be:

Correct Answer

📐 MA • Math Direct SubstitutionMathematicsAlgebra

Option C

Quick Summary:

Express the expression as $((a+b)^2 - 2ab) / ab$ . Just substitute the values directly into the simplified algebraic identity to get the answer in seconds.

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📐MAMath SolutionDirect Substitution

📋 Given

a + b = 12 and ab = 8

🔢 Formula Used

ab+ba=a2+b2ab=(a+b)2−2abab\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} = \frac{(a+b)^2 - 2ab}{ab}ba​+ab​=aba2+b2​=ab(a+b)2−2ab​

⚡ Exam Hall Shortcut / Speed Trick

Express the expression as $((a+b)^2 - 2ab) / ab$ . Just substitute the values directly into the simplified algebraic identity to get the answer in seconds.

⚠️ Common Student Trap / Pitfall

Students often incorrectly calculate $a^2 + b^2$ as $(a+b)^2$ , forgetting to subtract the $2ab$ term.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Simplify the expression

To solve for $\frac{a}{b} + \frac{b}{a}$ , find a common denominator:

ab+ba=a2+b2ab\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}ba​+ab​=aba2+b2​

Use algebraic identity

Recall the identity $a^2 + b^2 = (a+b)^2 - 2ab$ and substitute it into the expression:

a2+b2ab=(a+b)2−2abab\frac{a^2 + b^2}{ab} = \frac{(a+b)^2 - 2ab}{ab}aba2+b2​=ab(a+b)2−2ab​

Calculate the numerical value

Substitute $a+b = 12$ and $ab = 8$ into the formula:

(12)2−2(8)8=144−168=1288=16\frac{(12)^2 - 2(8)}{8} = \frac{144 - 16}{8} = \frac{128}{8} = 168(12)2−2(8)​=8144−16​=8128​=16

✅

C is correct because the substitution of given values $a+b=12$ and $ab=8$ into the identity $\frac{(a+b)^2 - 2ab}{ab}$ results in $16$ .

Core Concepts Used

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Algebraic Identities Rational Expressions Substitution Method

💡 EXAM TIP

This type of algebraic manipulation is essential for competitive exams in topics like Quadratic Equations and Simplification.

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