Question

Question:-
The sum of the squares of the digits of a two-digit number is 13.If we subtract 9 from that number , we get a number consisting of the same
digits written in the reverse order. Find the number.

Answer 1

Question:-

• The sum of the squares of the digits of a two-digit number is 13.If we subtract 9 from that number , we get a number consisting of the same digits written in the reverse order. Find the number.

To Find:-

• Fine the number.

Solution:-

Here ,

Let the numbers be x and y

Given ,

Sun of the squares of number is 13:-

• $\tt \: { x }^{ 2 } + { y }^{ 2 } = 13$

According to the Question:-

$\tt\implies \: 10x + y - 9 = 10y + x$

$\tt\implies \: 10x - x - 10y + y = 9$

$\tt\implies \: 9x - 9y = 9$

$\tt\implies \: 9( x - y ) = 9$

$\tt\implies \: x - y = \cancel\dfrac { 9 } { 9 }$

$\tt\implies \: x - y = 1 . . . . ( i )$

Now ,

$\tt\implies \: { ( x - y ) }^{ 2 } = { x }^{ 2 } + { y }^{ 2 } - 2xy$

$\tt\implies \: 1 = 13 - 2xy$

$\tt\implies \: 2xy = 12$

$\tt\implies \: xy = \cancel\dfrac { 12 } { 2 }$

$\tt\implies \: y = \dfrac { 6 } { x }$

Now ,

Substitute y in equation ( i ):-

$\tt\implies \: x - \dfrac { 6 } { x } = 1$

$\tt\implies \: { x }^{ 2 } - x - 6 = 0$

$\tt\implies \: { x }^{ 2 } + 2x - 3x - 6 = 0$

$\tt\implies \: x( x + 2 ) - 3( x + 2 ) = 0$

$\tt\implies \: ( x + 2 ) ( x - 3 ) = 0$

$\tt\implies \: x = - 2 , 3$

Hence ,

• The number is 32

Answer 2

Acçórding to the quéstion:-

⚘Súbtract 9 fróm the oríginál númber , we get a numbér consísting of the sáme digits writtén in the revérse órder.

$\sf{ \to \color{blue}óriginal\:number-9 = réverse\: fórm\:óf \: óriginàl\:númbér}$

$\sf{ \implies(10y+ a) - 9 = (10a +y)}$

Tránspósíng the líke térms

$\sf{ \implies10y -y -9 = 10a-a} \\ \\ \sf{ \implies9y -9 = 9a} \\ \\ \sf{ \implies9×(y-1 )= 9a} \\ \\ \bold{ \implies \red{y -1 = x}\: \: \: \: ....(2.)}$

Nów súbsitúte the válúe of x = ( y-1) in a ²+y² = 13

$\sf{ \implies a²+y² = 13} \\ \\ </p><p>\sf{ \implies(y-1)² +y² =13 }\\ \\ </p><p>\sf{ \implies y²+1 -2y +y² =13 }\\ \\ </p><p>\sf{ \implies 2y² -2y +1 =13 }\\ \\ </p><p>\sf{ \implies 2y² -2y = 12 }\\ \\</p><p>y²- y = 6$

míddlé splít the quàdratiç éqúatión

$\sf{ \implies y²+2y -3y -6 = 0 }\\ \\ \sf{ \implies y (y+2)-3(y+2) = 0} \\ \\ \sf{\implies(y+2)(y-3) = 0 }\\ \\ \sf{ \implies y = - 2 \: ór \: 3}$

⚘So the válue of y is 3

pút the valúe of y in eqúatíon (2)

$\sf{ \implies x = y-1 }\\ \\ \sf{ \implies a= 3-1 }\\ \\ \sf{ \implies \red{ a= 2}}$

⚘Theréfóre the númbér fórméd is 32

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