♣️ Distance formula :
[tex] \: \bf\implies \: \: d \: = \: \sqrt{ \bigg( x_{2} - x_{1} \bigg) ^{2} + \bigg( y_{2} - y_{1} \bigg) ^{2} } [/tex]
Given that,
The distance between the points A = (2,3) and B = (-1,y) is 5
According to the question by using formula we get,
[tex] \: \bf\implies \: \: d \: = \: \sqrt{ ( x_{2} - x_{1} ) ^{2} + ( y_{2} - y_{1} ) ^{2} } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ ( - 1 - 2) ^{2} + ( y - 3 ) ^{2} } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ ( - 3) ^{2} + ( y - 3 ) ^{2} } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ ( 9 ) + ( y ^{2} + 3 ^{2} - 6y ) } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{( 9 ) + ( y ^{2} + 9 - 6y ) } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ 9 + y ^{2} + 9 - 6y } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ y ^{2} - 6y + 9 + 9} [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ y ^{2} - 6y + 18} [/tex]
Squaring on both sides :
[tex]\: \sf\implies \: \: (5) ^{2} \: = \: \sqrt{ y ^{2} - 6y + 18} ^{2} [/tex]
[tex]\: \sf\implies \: \: 25\: = \: y ^{2} - 6y + 18[/tex]
[tex]\: \sf\implies \: \: 25 - 18\: = \: y ^{2} - 6y [/tex]
[tex]\: \sf\implies \: \: 7\: = \: y ^{2} - 6y [/tex]
[tex]\: \sf\implies \: \: 0\: = \: y ^{2} - 6y - 7[/tex]
[tex]\: \sf\implies \: \: 0\: = \: y ^{2} - 7y + y - 7[/tex]
[tex]\: \sf\implies \: \: ( y ^{2} - 7y ) \: \: (y - 7) \: = \: 0[/tex]
[tex]\: \sf\implies \: \:y ( y - 7 ) \: \: 1(y - 7) \: = \: 0[/tex]
[tex]\: \sf\implies \: \: ( y + 1) \: \: (y - 7) \: = \: 0[/tex]
[tex]\: \sf\implies \: \: ( y + 1) = 0 \: \: \: \: \: or \: \: \: \: \: (y - 7) = 0[/tex]
[tex]\: \sf\implies \: \: y + 1= 0 \: \: \: \: \: or \: \: \: \: \: y - 7 = 0[/tex]
[tex]\: \sf\implies \: \: y = - 1 \: \: \: \: \: or \: \: \: \: \: y = 7[/tex]
Hence,
[tex] \boxed{ \bf \: \therefore \: The \: \: value \: \: of \: \: y \: \: i s \: - 1 \: \: or \: \: 7 \: }[/tex]
[tex] \: [/tex]
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Given :-
To Find :-
Formula Used :-
♣️ Distance formula :
[tex] \: \bf\implies \: \: d \: = \: \sqrt{ \bigg( x_{2} - x_{1} \bigg) ^{2} + \bigg( y_{2} - y_{1} \bigg) ^{2} } [/tex]
Solution :-
Given that,
The distance between the points A = (2,3) and B = (-1,y) is 5
According to the question by using formula we get,
[tex] \: \bf\implies \: \: d \: = \: \sqrt{ ( x_{2} - x_{1} ) ^{2} + ( y_{2} - y_{1} ) ^{2} } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ ( - 1 - 2) ^{2} + ( y - 3 ) ^{2} } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ ( - 3) ^{2} + ( y - 3 ) ^{2} } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ ( 9 ) + ( y ^{2} + 3 ^{2} - 6y ) } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{( 9 ) + ( y ^{2} + 9 - 6y ) } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ 9 + y ^{2} + 9 - 6y } [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ y ^{2} - 6y + 9 + 9} [/tex]
[tex] \: \sf\implies \: \: 5 \: = \: \sqrt{ y ^{2} - 6y + 18} [/tex]
Squaring on both sides :
[tex]\: \sf\implies \: \: (5) ^{2} \: = \: \sqrt{ y ^{2} - 6y + 18} ^{2} [/tex]
[tex]\: \sf\implies \: \: 25\: = \: y ^{2} - 6y + 18[/tex]
[tex]\: \sf\implies \: \: 25 - 18\: = \: y ^{2} - 6y [/tex]
[tex]\: \sf\implies \: \: 7\: = \: y ^{2} - 6y [/tex]
[tex]\: \sf\implies \: \: 0\: = \: y ^{2} - 6y - 7[/tex]
[tex]\: \sf\implies \: \: 0\: = \: y ^{2} - 7y + y - 7[/tex]
[tex]\: \sf\implies \: \: ( y ^{2} - 7y ) \: \: (y - 7) \: = \: 0[/tex]
[tex]\: \sf\implies \: \:y ( y - 7 ) \: \: 1(y - 7) \: = \: 0[/tex]
[tex]\: \sf\implies \: \: ( y + 1) \: \: (y - 7) \: = \: 0[/tex]
[tex]\: \sf\implies \: \: ( y + 1) = 0 \: \: \: \: \: or \: \: \: \: \: (y - 7) = 0[/tex]
[tex]\: \sf\implies \: \: y + 1= 0 \: \: \: \: \: or \: \: \: \: \: y - 7 = 0[/tex]
[tex]\: \sf\implies \: \: y = - 1 \: \: \: \: \: or \: \: \: \: \: y = 7[/tex]
Hence,
[tex] \boxed{ \bf \: \therefore \: The \: \: value \: \: of \: \: y \: \: i s \: - 1 \: \: or \: \: 7 \: }[/tex]
[tex] \: [/tex]