Question

if the distance between the points (p -5)and (2 7) is 13 units hen the value of p is ?

Answer 1

$\: {\underline{\sf{\purple{\mathfrak{ Question }}}}}$

$☞︎︎︎ \: \mathsf{if \:the \: distance \: between \: the \: points \: (p \: , - 5) }$

$⠀⠀\mathsf{and(2,7) \: is \: 13 \: units \: then \: the \: value \: of \: p \: is \: ?}$

$\: {\underline{\sf{\purple{\mathfrak{ To \ Find }}}}}$

$☞︎︎︎ \: \mathsf{The \: value \: of \: p}$

$\: {\underline{\sf{\purple{\mathfrak{ Formula }}}}}$

$\mathsf{Distance \: Formula = }\mathtt{ \sqrt{(x{_2}- x {_1}) ^{2} +(y{_2} - y{_1} )^{2} }}$

$\: {\underline{\sf{\purple{\mathfrak{ Given }}}}}$

$☞︎︎︎ \: \: \: \: \: \: \mathtt{x{_1}=p \: \: \: \: \: \: \: \: \: \: \: x{_2}=2}$

$☞︎︎︎ \: \: \: \: \: \mathsf{y{_1}= - 5 \: \: \: \: \: \: \: \: \: \: y{_2} = 7}$

$\: {\underline{\sf{\purple{\mathfrak{ Solution }}}}}$

$☞︎︎︎ \: \mathsf{By \: Using \: Distance \: Formula}$

$\implies \: \: \mathtt{D = \sqrt{(x{_2}- x {_1}) ^{2} +(y{_2} - y{_1} )^{2} }}$

$\implies\: \: \mathsf{13 = { \mathsf{ \sqrt{[ \: 2-p \: ]^{2} +[ \: 7-(-5)\: ]^{2} }}} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \mathsf{Squaring \: Both \: Sides}$

$\implies \: \: \mathsf{(13)^{2} = { \mathsf({ \sqrt{[ \: 2-p \: ]^{2} +[ \: 7-(-5)\: ]^{2} }}} }) ^{2}$

$\implies \: \: \mathsf{169 = { \mathsf({{[ \: 2-p \: ]^{2} +[ \: 7-(-5)\: ]^{2} }}} }$

$\implies \: \: \mathsf{169 = { \mathsf({{[ {2}^{2} + 2 (2)(p) + {p}^{2} ] +[ \: 12 ]^{2} }}} }$

$\implies \: \: \mathsf{169 = { \mathsf({{[ 4 + 4p+ {p}^{2} ] +144 }}} }$

$\implies \: \: \mathsf{169 = { \mathsf{{{p}^{2} + 4p +148}}} }$

$\implies \: \: \mathsf{ { \mathsf{{{p}^{2} + 4p +148 - 169 = 0}}} }$

$\implies \: \: \mathsf{ { \mathsf{{{p}^{2} + 4p - 21= 0}}} }$

$\implies \: \: \mathsf{ { \mathsf{{{p}^{2} - 3p + 7p - 21= 0}}} }$

$\implies \: \: \mathsf{ { \mathsf{{p(p - 3) + 7(p - 3)= 0}}} }$

$\implies \: \: \mathsf{p-3=0 \: \: \: \: OR \: \: \: \: \: p+7=0 }$

$\implies \: \: \mathsf{p = 3 \: \: \: \: OR \: \: \: \: \: p = - 7 }$

$\mathsf{\red{ \boxed{\mathsf{{Answer }}}}:- \: \boxed{{\mathsf { {p = 3}} }} \: \: \: \: OR \: \: \: \: \: \boxed{{{ {\mathsf p = - 7 }}}}}$

Answer 2

$\LARGE\color{goldenrod}{★}$ $\LARGE\bf{\color{teal}{ɢɨʋɛռ:-}}$

$\hookrightarrow$ $\mathtt{x_1\:=\:p}$

$\hookrightarrow$ $\mathtt{x_2\:=\:2}$

$\hookrightarrow$ $\mathtt{y_1\:=\:-5}$

$\hookrightarrow$ $\mathtt{y_2\:=\:7}$

$\LARGE\color{goldenrod}{★}$ $\LARGE\bf{\color{teal}{ȶօʄɨռɖ:-}}$

$\hookrightarrow$ $\mathtt{The\:value\:of\:p}$

$\LARGE\color{goldenrod}{★}$ $\LARGE\bf{\color{teal}{աɛӄռօա:-}}$

$\odot\mid$ $\large\mathtt{\color{maroon}{Distance\: Formula::}}$ $\rightarrow$

$\:$

$\large\boxed{\bf{\sqrt{(x_2-x_1)²+(y_2-y_1)²}}}$

$\:$

$\LARGE\color{goldenrod}{★}$ $\LARGE\bf{\color{teal}{ֆօʟʊȶɨօռ:-}}$

$\leadsto$ $\large\mathtt{\color{maroon}{By\:using\: Formula::}}$

$\implies\mathtt{D={\bf{\sqrt{(x_2-x_1)²+(y_2-y_1)²}}}}$

$\implies\mathtt{13={\bf{\sqrt{[2-p]²+[7-(-5)]²}}}}$

$\leadsto$ $\large\mathtt{\color{maroon}{By\: squaring\:both\:sides::}}$

$\implies\mathtt{(13)²=({\bf{\sqrt{[2-p]²+[7-(-5)]²)²}}}}$

$\implies\mathtt{169=[2²+2(2)(p)+p²]+(12)²}$

$\implies\mathtt{169=[4+4p+p²]+144}$

$\implies\mathtt{169=p²+4p+148}$

$\implies\mathtt{p²+4p+148-169=0}$

$\implies\mathtt{p²+4p-21=0}$

$\implies\mathtt{p(p-3)+7(p-3)=0}$

$\implies\mathtt{p-3=0\:\:or\:\:p+7=0}$

$\implies\mathtt{\color{maroon}{p=3\:\:or\:\:p=-7}}$

$\:$

$\large\color{teal}{ \underline{ \boxed{ \odot \mid{ \bf{p=3\:\:or\:\:p=-7}}}}}$