Math, asked by pardeep143pk64, 9 months ago

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

Answers

Answered by ItzAditt007
3

Answer:-

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

  1. 2 and,
  2. - 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.

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