Question

If the difference of the squares of two numbers is 45,
the square of the smaller number is 4 times the larger
number, then the rumber are :
(A) 9, 6 or 9, -6
(B) 5, 6 or 5, -6
(C) 9,5 or 9, -5
(D) None of these​

Answer 1

Given :

• Difference of squares of two numbers = 45
• Square of smaller number is 4 times the larger number

To Find :

• The numbers

Solution :

Let the two numbers be "x" and "y"

Then , According to the given conditions ;

• x² - y² = 45 .............(1)
• y² = 4x ............(2)

Substituting the value of y² in equation(1) ,

$\\ : \implies \sf \: {x}^{2} - 4x - 45 = 0$

$\\ : \implies \sf \: {x}^{2} - 9x + 5x - 45 = 0$

$\\ : \implies \sf \: x(x - 9) + 5(x - 9) = 0$

$\\ : \implies \sf \: (x - 9)(x + 5) = 0$

$\\ : \implies{ \underline{\boxed{\pink{\mathfrak{ \: x = 9 \: (or) \: x = - 5}}}}} \: \bigstar$

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Substituting x value as 9 in equation(1) ,

$\\ : \implies \sf \: {(9)}^{2} - {y}^{2} = 45$

$\\ : \implies \sf \: 81 - {y}^{2} = 45$

$\\ : \implies \sf \: - {y}^{2} = 45 - 81$

$\\ : \implies \sf \: - {y}^{2} = - 36$

$\\ : \implies \sf \: {y}^{2} = 36$

$\\ : \implies \sf \: y = \sqrt{36}$

$\\ : \implies{\underline{\boxed{\pink{\sf{ \: y = \pm \: 6}}}}} \: \bigstar$

Now , Substituting x as -5 in equation(1) ;

$\\ : \implies \sf \: {( - 5)}^{2} - {y}^{2} = 45$

$\\ : \implies \sf \: 25 - {y}^{2} = 45$

$\\ : \implies \sf \: - {y}^{2} = 45 - 25$

$\\ : \implies \sf - {y}^{2} = 20$

$\\ : \implies \sf \: {y}^{2} = -20$

Square of a number can't be negative. So , ignoring the value we got then ,

• The values of x and y are 9 , 6 (or) 9 , -6. Hence , Option(A) is the required answer

Answer 2

$\huge\fcolorbox{black}{lime}{Answer}$

Let the two numbers be a and b

a2−b2=45.............................(1)

 b2=4a..............................................(2)

Putting (2) in (1)

a2−4a=45

a2−4a−45=0

a2−9a+5a−45=0

 a(a−9)+5(a−9)=0

(a−9)(a+5)=0

a=9ora=−5

If a=5,  b=25−b2=45

                 25−45=b2

                  b2=20(Neglecting this value)

IF a=9   81−b2=45

               81−45=b2

                b2=36

                 b=±6

Numbers are 9,6 or 9,-6