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Q.

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. What are the numbers?


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a

±12 and 18  

b

±12 and 20  

c

±2 and 18  

d

±13 and 18   

answer is A.

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Detailed Solution

Given that the difference of squares of two numbers is 180.
Also, the square of the smaller number is 8 times the larger number.
Let the smaller number be n.
Let the larger number be m.
Square of smaller number = n 2  .
Also, the square of smaller number is 8 times the larger number,
⇒ n 2 =8m  
Square of larger numberQuestion Image.
The difference of squares of numbers is 180.
So,
⇒ m 2 − n 2 =180 ⇒ m 2 − n 2 −180=0  
We know that,
n 2 =8m  .
Now, using the above relation,
m 2 −8m−180=0  .
Solve the above quadratic equation using factorisation method.
m 2 −8m−180=0 ⇒ m 2 −18m+10m−180=0  
Take out common terms.
⇒m(m−18)+10(m−18)=0 ⇒(m−18)(m+10)=0 ⇒m−18 or m+10=0 ⇒m=18 or m=−10  
As the larger number cannot be negative, m = 18.
Smaller number,
n 2 =8m  
⇒ n 2 =8×18 ⇒ n 2 =144  
Take square root on both sides,
⇒n= 144 ⇒n=±12  
Smaller number, n= ±12  .
Larger number, m=18.
The numbers can be 18 and 12 or 18 and -12.
Therefore,  the correct option is 1.
 
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