Question

if the distance between the point (5,-2) and
(1, a) is 5, find the value of a​

Answer 1

Please mark as brailiest

Answer 2

$\LARGE\textsf{\underline\textcolor{aqua}{↭ SoLuTioN :-}}$

$\large\textsf{ }$

↦ Distance Formula :-

$\sqrt{( \large\textsf{x}_{2} \: - \: \large\textsf{x}_{1} )^{2} + \: ( \large\textsf{y}_{2} \: - \: \large\textsf{y}_{1})^{2} }$

$\large\textsf{ }$

Here ,

↦ x₁ = 5

↦ y₁ = − 2

↦ x₂ = 1

↦ y₂ = a

↦ d = 5

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{5} \: = \: \sqrt{\large\textsf{( \: 1 \: - \: 5 \: })^{2} \: + \: [ \large\textsf{\: a \: - \: ( \: - 1 \: )}] ^{2} }$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{5} \: = \: \sqrt{ {\large\textsf{4}}^{2} \: + \: \large\textsf{( \: a \: + \: 1 \: )} ^{2} }$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{25} \: = \:\large\textsf{ 16 }\: + \: \: {\large\textsf{a}}^{2} \: + \large\textsf{ \: 1 \: - \: 2a}\large\textsf{.... ( Squaring on both sides )}$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{25} \: = \:\large\textsf{ 17 \: + \: {a}}^{2} \: - \: 2a$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{25 \: - \: 17 \:} = \: {\large\textsf{a}}^{2} \: - \large\textsf{ \: 2a }$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{8 \: = \: {a}}^{2} \: - \large\textsf{ \: 2a}$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow - \: {\large\textsf{a}}^{2} \: + \: \large\textsf{2a \: + 8 \: = \: 0 }$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow - \: {\large\textsf{a}}^{2} \: \large\textsf{ + 4a \: - \: 2a \: + 8 \: = \: 0}$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow \large\textsf{- \: a \: ( \: a \: - \: 4 \: ) \: - \: 2 \: ( \: a \: - \: 4 \: ) \: = \: 0}$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{( \: a \: - \: 4 \: )( \: a \: - \: 2 \:) = \: 0}$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{a \: - \: 4 \: = \: 0 \: \: or \: \: a \: - \: 2 \: = \: 0}$

$\large\textsf{ }$

$\qquad\tt{:}\longrightarrow\large\textsf{ a \: = \: 4 \: \: or \: \: a \: = \: 2}$

$\large\textsf{ }$

$\boxed{\large\textsf{∴ The Value of a = 4 or 2}}$