Question

If the distance between the points P(a, 7) and Q (1,3)
is 5 units, then the value of a is
options are
a)4 or 2
b) -2or4
c)-4or-2
d)2or-4
$ans \: with \: explanation \: plz$
​

Answer 1

Step-by-step explanation:

By distance formula

$(a - 1) {}^{2} + (7 - 3) {}^{2} = (5) {}^{2}$

$a = - 2 = 4$

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

• The distance between the points P(a, 7) and Q (1,3) is 5 units.

We know,

Distance Formula :-

Let us consider a line segment joining the points P and Q, then distance between P and Q is given by

$\bf\longmapsto \: PQ= \sqrt{ {(x_2-x_1)}^{2} + {(y_2-y_1)}^{2} }$

$\sf \: where \: coordinates \: are \:P (x_1,y_1) \: and \: Q(x_2,y_2)$

Here,

$\rm :\longmapsto\:x_1 = a, \: y_1 =7 , \: x_2= 1, \: y_2=3$

and

$\rm :\longmapsto\:PQ = 5$

On Substituting all these values, in above formula, we get

$\rm :\longmapsto\:5 = \sqrt{ {(1 - a)}^{2} + {(3 - 7)}^{2} }$

$\rm :\longmapsto\:5 = \sqrt{ {(1 - a)}^{2} + {( - 4)}^{2} }$

$\rm :\longmapsto\:5 = \sqrt{ {(1 - a)}^{2} + 16 }$

On squaring both sides, we get

$\rm :\longmapsto\: {5}^{2} = {(1 - a)}^{2} + 16$

$\rm :\longmapsto\: {25} = {(1 - a)}^{2} + 16$

$\rm :\longmapsto\: {25} - 16 = {(1 - a)}^{2}$

$\rm :\longmapsto\: {9} = {(1 - a)}^{2}$

$\rm :\longmapsto\: { {3}^{2} } = {(1 - a)}^{2}$

$\rm :\implies\:1 - a \: = \: \pm \: 3$

$\rm :\implies\:1 - a = 3 \: \: \: or \: \: \: 1 - a = - 3$

$\bf\implies \:a \: = \: - 2 \: \: \: \: or \: \: \: \: a \: = \: 4$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf{ \: Option \: (b) \: is \: correct}}}$

Additional Information :-

Section Formula

Let us consider a line segment joining the points A and B and let C (x, y) divides the line segment joining A B in the ratio m : n internally, then coordinates of C is

$\rm :\longmapsto\:(x, y) = \: \bigg(\dfrac{mx_2 + nx_1}{m + n} ,\dfrac{my_2 + ny_1}{m + n} \bigg)$

Midpoint Formula :-

Let us consider a line segment joining the points A and B and let C (x,y) be the midpoint of AB, then coordinates of C is given by

$\rm :\longmapsto\:(x, y) = \: \bigg(\dfrac{x_2 + x_1}{2} ,\dfrac{y_2 + y_1}{2} \bigg)$