Question

Find the value of 'a' when the distance between the point (3,a) and (4,1) is root over 10.

Answer 1

Answer:

Let, the points be P(3,a) and Q(4,1)

$PQ = \sqrt{(4 - 3) {}^{2} + (1 - a) {}^{2} }$

$\sqrt{10} = \sqrt{1 {}^{2} + (1 - a) {}^{2} }$

$10 = {1}^{2} + 1 + {a}^{2} - 2a$

$10 = 1 + 1 + {a}^{2} - 2a$

$10 = 2 + {a}^{2} - 2a$

${a}^{2} - 2a - 8 = 0$

${a}^{2} - (4 - 2)a - 8 = 0$

${a}^{2} - 4a + 2a - 8 = 0$

$a(a - 4) + 2(a - 4) = 0$

(a-4)(a+2)=0

Either,

a-4=0

a=4

Or,

a+2=0

a=-2

Therefore the value of a is 4 and - 2

Answer 2

$\footnotesize \tt \red{ Given:-}$

$\: \: \: \: \: \: \: \: \: \: \: \: \:$

$\red\bullet\footnotesize \tt { \: Distance = \sqrt{10} }$

$\red\bullet \footnotesize \tt { \: A = (3,a)}$

$\red\bullet\footnotesize \tt { \: B = (4,1)}$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt \red{ Solution:-}$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\red\bullet\footnotesize \tt { \: x2 = 4}$

$\red\bullet\footnotesize \tt { \: x1 = 3}$

$\red\bullet\footnotesize \tt { \: y2 = 1}$

$\red\bullet\footnotesize \tt { \: y1 = a}$

$\: \: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt{Distance \: (AB) = \: \sqrt{(x2 - x1)^{2} + (y2 - y1)^{2}}}$

$\footnotesize \tt{ \sqrt{10} = \: \sqrt{(4 - 3)^{2} + (1 - a)^{2}} }$

$\footnotesize \tt{ \sqrt{10} = \: \sqrt{(1)^{2} + (1) ^{2} - 2a+ a^{2}} }$

$\footnotesize \tt{ \sqrt{10} = \: \sqrt{1 + 1 - 2a+ a^{2}} }$

$\footnotesize \tt{ \sqrt{10} = \: \sqrt{2 - 2a+ a^{2}} }$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt \red{ Squaring \: both \: sides }$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt{ (\sqrt{10})^{2} = \: \sqrt{(2 - 2a+ a^{2}})^{2} }$

$\footnotesize \tt{ 10 = \: {2 - 2a+ a^{2}} }$

$\footnotesize \tt{ - {a}^{2} + 2a = 2 - 10 }$

$\footnotesize \tt{ - {a}^{2} + 2a = - 8 }$

$\: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt \red{ Now \: divide \: it \: by \: (-1) }$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt{ {a}^{2} - 2a = 8 }$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt \red{ Now \: factorise \: it \: by \: middle \: term \: splitting }$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt{ {a}^{2} - 2a - 8 =0 }$

$\footnotesize \tt{ {a}^{2} - 4a + 2a- 8 =0 }$

$\footnotesize \tt{ a(a - 4) + 2(a- 4)=0 }$

$\footnotesize \tt{ (a + 2)(a - 4)=0 }$

$\: \: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt{ a + 2=0 }$

$\boxed{ \underline{ \underline{ \footnotesize \tt \red \bullet{ \: a = - 2}}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt{ a - 4=0 }$

$\boxed{ \underline{ \underline{\footnotesize \tt \red \bullet{ \: a = 4 }}}}$

$\: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize \tt \red{ Value \: of \: a \: can \: be \: -2 \: or \: 4}$