Question

If x - 2y = 3 and xy = 5, find the value of $x^2-4y^2$

• A. 21
• B. 20
• C. 23
• D. 22

Answer 1

Answer:

A. 21

Given:

x - 2y = 3 and xy = 5

Formula:

(x - y)² = x² + y² - 2xy 

Calculation:

(x - 2y)² = x² + 4y² - $2\times{x}\times{2y}$

⇒ 3² = x² + 4y² - $4\times{5}$

⇒ x² + 4y² = 9 + 20 

⇒ x² + 4y² = 29     ...1)

(x² + 4y²)² = x⁴ + 16y⁴ + $2\times{x^2}\times{4y^2}$

⇒ $29^2=x^4+16y^4+8\times{25}$

⇒ $x^4+16y^4=841-200$ 

⇒ $x^4+16y^4=641$     ................2)

$(x^2-4y^2)^2=x^4+16y^4+2\times{x^2}\times{4y^2}$

⇒ $(x^2-4y^2)^2=641-8\times{25}$

⇒$(x^2-4y^2)^2=441$

⇒$x^2-4y^2=\sqrt{441}$

∴ $x^2-4y^2=21$.

Hence the answer is 21.

Shortcut Method:

By using substitution method 

if xy = 5 then the possibilities

 (x,y) = (1,5) (5,1)

 substitute the values x and y in the equation and check

 x - 2y = 5 - 2(1) = 3

x = 5, y = 1 satisfies 

substitute the values in this equaltion

$x^2-4y^2$=5²-4(1)²

$=25-4=21$

∴$x^2-4y^2$ is 21