Find the value of a, if the distance between the two points is: (4,-5)

1.	Find the value of a, if the distance between the two points is: (4,-5), (-2,a) is √85 units?
Answer» Answer:- There Are Two values of 'a' is possible:- 2 and, - 12. Explanation:- Given:- Two POINTS (4, -5) and (-2, a). Distance between these two points is $\tt\sqrt{85}$ Units. To FIND:- The possible values of a. Formula USED:- Distance Formula, $\gray{\bullet \boxed{ \green{ \bf d = \sqrt{(x_2 - x_1) {}^{2} + (y_2 - y_1) {}^{2}. }}}}$ Where, $\tt x_2\:\:and\:\:x_2$ Are the x coordinates of both the points. $\tt y_1\:\:and\:\:y_2$ Are the y coordinates of both the points. d = Distance between both the points. So Here, $\tt x_2\:\:and\:\:x_2$ Are -2 and 4 respectively. $\tt y_2\:\:and\:\:y_2$ Are a and -5 respectively. d = $\sqrt{85}$ Units. Now, By PUTTING the above values in formula we get, $\\ \bf\mapsto \sqrt{(x_2 - x_1) {}^{2} + (y_2 - y_1) {}^{2}} = d. \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt \mapsto \sqrt{ \bigg(( - 2) - (4) \bigg) {}^{2} + \bigg( ( a)- ( - 5) \bigg) {}^{2}} = \sqrt{85} \: units. \\ \\ \tt\mapsto \sqrt{( - 2 - 4) {}^{2} + (a + 5) {}^{2} } = \sqrt{85} \: units. \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt\mapsto \sqrt{ (- 6) {}^{2} + ( {a}^{2} + 2 \times a \times 5 + 5 {}^{2} ) } = \sqrt{85} \: units \: \: \: \: \: \: \: \: \: \: \:$ $\\ \tt\mapsto\sqrt{ {a}^{2} + 10a + 61} = \sqrt{85} \: units. \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt\mapsto \bigg(\sqrt{ {a}^{2} + 10a + 61} \bigg) {}^{2} = \bigg( \sqrt{85} \bigg) {}^{2} units. \\ \\ \bf(squaring \: \: both \: \: the \: \: sides).$ $\\ \tt\mapsto {a}^{2} + 10a + 61 =85. \: \: \: \: \: \: \: \: \: \\ \\ \tt\mapsto {a}^{2} + 10a + 61 - 85 = 0. \\ \\ \tt\mapsto {a}^{2} + 10a - 24 = 0. \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt\mapsto {a}^{2} + (12 - 2)a - 24 = 0. \\ \\ \bf(splitting \: \: middle \: \: term) \\ \\ \tt\mapsto {a}^{2} + 12a - 2a - 24 = 0. \\ \\ \tt\mapsto a(a + 12) - 2(a + 12) = 0. \\ \\ \tt\mapsto(a + 12)(a - 2) = 0. \: \: \: \: \: \: \: \: \: \: \: \: \: \:$ $\\ \tt\mapsto either \: (a + 12) = 0 \: \: or \: \: (a - 2) = 0. \\ \\ \tt \red{\mapsto \: \: \boxed{ \blue{ \bf either \: \: a = - 12 \: \: \: \: or \: \: \: \: a = 2.}}}$ Therefore The required values of a are -12 and 2.

Find the value of a, if the distance between the two points is: (4,-5), (-2,a) is √85 units?

Answer»

Answer:-

There Are Two values of 'a' is possible:-

2 and,
- 12.

Explanation:-

Given:-

Two POINTS (4, -5) and (-2, a).

Distance between these two points is $\tt\sqrt{85}$ Units.

To FIND:-

The possible values of a.

Formula USED:-

Distance Formula,

$\gray{\bullet \boxed{ \green{ \bf d = \sqrt{(x_2 - x_1) {}^{2} + (y_2 - y_1) {}^{2}. }}}}$

Where,

$\tt x_2\:\:and\:\:x_2$ Are the x coordinates of both the points.

$\tt y_1\:\:and\:\:y_2$ Are the y coordinates of both the points.

d = Distance between both the points.

So Here,

$\tt x_2\:\:and\:\:x_2$ Are -2 and 4 respectively.

$\tt y_2\:\:and\:\:y_2$ Are a and -5 respectively.

d = $\sqrt{85}$ Units.

Now,

By PUTTING the above values in formula we get,

$\\ \bf\mapsto \sqrt{(x_2 - x_1) {}^{2} + (y_2 - y_1) {}^{2}} = d. \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt \mapsto \sqrt{ \bigg(( - 2) - (4) \bigg) {}^{2} + \bigg( ( a)- ( - 5) \bigg) {}^{2}} = \sqrt{85} \: units. \\ \\ \tt\mapsto \sqrt{( - 2 - 4) {}^{2} + (a + 5) {}^{2} } = \sqrt{85} \: units. \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt\mapsto \sqrt{ (- 6) {}^{2} + ( {a}^{2} + 2 \times a \times 5 + 5 {}^{2} ) } = \sqrt{85} \: units \: \: \: \: \: \: \: \: \: \: \:$

$\\ \tt\mapsto\sqrt{ {a}^{2} + 10a + 61} = \sqrt{85} \: units. \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt\mapsto \bigg(\sqrt{ {a}^{2} + 10a + 61} \bigg) {}^{2} = \bigg( \sqrt{85} \bigg) {}^{2} units. \\ \\ \bf(squaring \: \: both \: \: the \: \: sides).$

$\\ \tt\mapsto {a}^{2} + 10a + 61 =85. \: \: \: \: \: \: \: \: \: \\ \\ \tt\mapsto {a}^{2} + 10a + 61 - 85 = 0. \\ \\ \tt\mapsto {a}^{2} + 10a - 24 = 0. \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \tt\mapsto {a}^{2} + (12 - 2)a - 24 = 0. \\ \\ \bf(splitting \: \: middle \: \: term) \\ \\ \tt\mapsto {a}^{2} + 12a - 2a - 24 = 0. \\ \\ \tt\mapsto a(a + 12) - 2(a + 12) = 0. \\ \\ \tt\mapsto(a + 12)(a - 2) = 0. \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\\ \tt\mapsto either \: (a + 12) = 0 \: \: or \: \: (a - 2) = 0. \\ \\ \tt \red{\mapsto \: \: \boxed{ \blue{ \bf either \: \: a = - 12 \: \: \: \: or \: \: \: \: a = 2.}}}$

Therefore The required values of a are -12 and 2.