Question

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. The two numbers are $18,12$ or $18,-12$

• A. True
• B. False

Answer 1

The correct option is A True

$\Rightarrow$ Let the large and smaller number be $x$ and $y$ respectively.

According to the question,

$\Rightarrow$ $x^2-y^2=180$ and $y^2=8x$

$\Rightarrow$ $x^2-8x=180$

$\Rightarrow$ $x^2-8x-180=0$

$\Rightarrow$ $x^2-18x+10x-180=0$

$\Rightarrow$ $x(x-18)+10(x-18)=0$

$\Rightarrow$ $(x-18)(x+10)=0$

The larger number cannot be negative as $8$ times of larger number will be negative and hence, the square of the smaller number will be negative which is not possible.

$\therefore$ The larger number will be $18$ only.

$\Rightarrow$ $x=18$

$\Rightarrow$ $y^2=8x=8\times 18=144$

$\therefore$ $y=\pm 12$

$\therefore$ Smaller numbers $=\pm 12$

$\therefore$ The numbers are $18$ and $12$ or $18$ and $-12$