Question

Find all possible values of a when the distance between: (a,-1 )and (5,3) is 5 units

Answer 1

Given ,

• The distance between (a,-1 )and (5,3) is 5 units.

To find:—

• The value of a .

★ Applying the distance formula we get ,

$\color{red} \bold{ \sqrt{ {(5 - a)}^{2} + (3 - ( - 1)) ^{2} } = 5}$

$\color{blue} \bold{ \implies\sqrt{ {(5 - a)}^{2} + (3 + 1) ^{2} } = 5}$

$\color{blue} \bold{ \implies\sqrt{ {(5 - a)}^{2} + 4 ^{2} } = 5}$

$\color{blue} \bold{ \implies\sqrt{ 25 - 10a + {a}^{2} + 16 } = 5}$

$\color{blue} \bold{ \implies(\sqrt{{a}^{2} - 10a+ 41} )^{2} = {5}^{2} }$

$\color{blue} \bold{ \implies{a}^{2} - 10a+ 41 = 25 }$

$\color{blue} \bold{ \implies{a}^{2} - 10a+ 16 =0 }$

$\color{blue} \bold{ \implies{a}^{2} - 8a - 2a+ 16 =0 }$

$\color{blue} \bold{ \implies \: a(a - 8) - 2(a - 8)= }$

$\color{blue} \bold{ \implies \: (a - 8)(a - 2)= }$

$\color{blue} \bold{ \implies \: a - 8 = 0 \: \: \: \: \: or \: \: \: a - 2= 0}$

$\color{blue} \bold{ \implies \: a = 8 \: \: \: \: \: or \: \: \: a = 2}$

So,

★ The value of a is 8 and 2