Question : If $x^2+8 y^2+12 y-4 x y+9=0$, then the value of $(7 x+8 y)$ is:
Option 1: –33
Option 2: 9
Option 3: 33
Option 4: –9
Correct Answer: 9
Solution :
Given: $x^2+ 8y^2- 12y - 4xy + 9 = 0$
⇒ $x^2+ 4y^2+ 4y^2- 12y - 4xy + 9 = 0$
⇒ $(x^2+ 4y^2- 4xy) + (4y^2- 12y + 9) = 0$
⇒ $(x - 2y)^2+ (2y - 3)^2= 0$
If the sum of squares of two numbers is zero then each number will also be zero.
⇒ $(2y - 3)^2= 0$ and $(x - 2y)^2= 0$
⇒ $(2y - 3) = 0$
⇒ $2y = 3$
⇒ $y = \frac{3}{2}$
and $(x - 2y)^2= 0$
⇒ $(x - 2y) = 0$
⇒ $x = 2× \frac{3}{2}$
⇒ $x = 3$
$\therefore (7x - 8y) = 7 × 3 - 8 × \frac{3}{2}= 21 - 12=9$
Hence, the correct answer is 9.
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