Question

9)If the distance between the points (8,p) and (4,3) is 5 units, then the value of p is *

Answer 1

$\bf \large \underline{ \purple{GIVEN}}$

• Given points: (8,p) and (4,3)
• Distance between the points is 5 units.

$\: \bf \large \underline{ \purple{SOLUTION}}$

➸ Distance formula:

$\to {\blue{ \sf\sqrt{( x_{2} -x_{1}) {}^{2} + (y_{2} - y_{1} ) {}^{2} } }}$

➸We need to find the value of p,

ㅤㅤㅤ☛By using distance formula,

$\sf\sqrt{( 4 -8) {}^{2} + (3 - p ) {}^{2} } = 5$

$\sf \sqrt{ ( - 4) {}^{2} + 9 + p {}^{2} - 6p } = 5.....( \because \: a {}^{2} - b {}^{2} = a {}^{2} + b {}^{2} - 2ab)$

$\sf \sqrt{16 + 9 + p {}^{2} - 6p} = 5$

$\sf( \sqrt{25 + p {}^{2} - 6p } ) {}^{2}= (5) {}^{2}.....( \: squaring \: both \: sides)$

$\sf \: p {}^{2} - 6p + 25 = 25$

$\sf \: p {}^{2} - 6p = 0$

$\sf \: p(p - 6) = 0.......(eq.1)$

__________________

➸From equation 1,

$\bf \: \: \: \: \: \: p = 0$

$\bf \: \: \: \: \: \: p - 6 = 0 \to \: p = 6$

ㅤ

Hence,

$\implies \bf \boxed{p = 0} \: or \: \boxed{\: p = 6}$

Answer 2

Given ,

The distance between the points (8,p) and (4,3) is 5 units

We know that ,

The distance between two points is given by

$\boxed{ \sf{D = \sqrt{ {( x_{2} - x_{1} )}^{2} + {(y_{2} - y_{1} )}^{2} } }}$

Thus ,

$\sf \mapsto 5 = \sqrt{ {(4 - 8)}^{2} + {(3 - p)}^{2} } \\ \\ \sf Squaring \: on \: both \: sides \: , \: we \: get \\ \\\sf \mapsto 25 = 16 + 9 + {(p)}^{2} - 6p \\ \\ \sf \mapsto {(p)}^{2} - 6p + 25 - 25 = 0 \\ \\ \sf \mapsto {(p)}^{2} - 6p = 0 \\ \\ \sf \mapsto p(p - 6) = 0 \\ \\ \sf \mapsto p = 0 \: \: or \: \: p = 6$

Therefore ,

The value of p will be 0 or 6