If $x + y = 10$ and $xy = 16$, then the numerical value of $\l... | Mathematics MCQ | Examoogle

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If $x + y = 10$ and $xy = 16$ , then the numerical value of $\left( \frac{x}{y} + \frac{y}{x} \right)$ will be:

1.25

2.125

3.5

4.25

Correct Answer

📐 MA • Math Algebraic IdentityMathematicsAlgebra

Option B

125

Quick Summary:

Express the expression as $\frac{(x+y)^2 - 2xy}{xy}$ . Simply substitute the sum (10) and product (16) directly into the values: $\frac{100 - 32}{16} = \frac{68}{16} = 4.25$ .

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📐MAMath SolutionAlgebraic Identity

📋 Given

Sum of variables x + y = 10 and their product xy = 16.

🔢 Formula Used

xy+yx=x2+y2xy=(x+y)2−2xyxy\frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy} = \frac{(x + y)^2 - 2xy}{xy}yx​+xy​=xyx2+y2​=xy(x+y)2−2xy​

⚡ Exam Hall Shortcut / Speed Trick

Express the expression as $\frac{(x+y)^2 - 2xy}{xy}$ . Simply substitute the sum (10) and product (16) directly into the values: $\frac{100 - 32}{16} = \frac{68}{16} = 4.25$ .

⚠️ Common Student Trap / Pitfall

Students often try to solve for $x$ and $y$ individually using the quadratic equation $t^2 - 10t + 16 = 0$ , which is unnecessary and time-consuming.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Simplify the expression

Find a common denominator for the given expression $\frac{x}{y} + \frac{y}{x}$ .

xy+yx=x2+y2xy\frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy}yx​+xy​=xyx2+y2​

Apply algebraic identities

Use the identity $x^2 + y^2 = (x + y)^2 - 2xy$ to express the numerator in terms of given values.

(x+y)2−2xyxy\frac{(x + y)^2 - 2xy}{xy}xy(x+y)2−2xy​

Substitute given values

Substitute $x+y = 10$ and $xy = 16$ into the formula.

(10)2−2(16)16=100−3216\frac{(10)^2 - 2(16)}{16} = \frac{100 - 32}{16}16(10)2−2(16)​=16100−32​

Final calculation

Calculate the final numerical value by dividing the result.

6816=4.25\frac{68}{16} = 4.251668​=4.25

✅

D is correct because the substitution of the given values into the algebraic identity results in 4.25.

Core Concepts Used

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Algebraic Identities Quadratic Relationships Fraction Simplification

💡 EXAM TIP

This identity $(x+y)^2 = x^2 + 2xy + y^2$ is fundamental for solving problems involving symmetric polynomials in both Algebra and Coordinate Geometry.

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