Question

Find the value of k if p(4 -2) is the mid point of line segment joining the points A(5k,3) and B(-k,-7)

Answer 1

Answer:-

Given:

P (4 , - 2) is the midpoint of the line segment joining the points A(5k , 3) & B( - k , - 7).

We know that midpoint of a line segment joining the points (x₁ , y₁) & (x₂ , y₂) is :

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

Let,

• x = 4

• y = - 2

• x₁ = 5k

• x₂ = - k

• y₁ = 3

• y₂ = - 7

Putting the values we get,

$\implies \sf \: (4 \: \: ,\: \: - 2) = \bigg( \dfrac{5k - k}{2} \: \: ,\: \: \dfrac{3 - 7}{2} \bigg) \\ \\ \implies \sf \: (4 \: \:, \: \: - 2) = \bigg( \dfrac{4k}{2} \: \: , \: \: \dfrac{ - 4}{2} \bigg) \\ \\ \implies \sf \:4 = \frac{4k}{2} \\ \\ \implies \sf \:4 \times 2 = 4k \\ \\ \implies \boxed{ \sf \:2 = k}$

∴ The value of k is 2.

Answer 2

Answer:

Given :-

➙The coordinates of the points are :

$\mapsto$ A(5k , 3), B(-k , -7) and P(4 , -2)

$\leadsto$ We know that, the mid point formula between two points is given by,

➲ ($\dfrac{\sf{x_1} + {x_2}}{2}$ , $\dfrac{\sf{y_1} + {y_2}}{2}$)

$\Rightarrow$ We can equate the above formula with the coordinates of the mid point, P to get the value of k.

➣ According to the question by using the formula we get,

⇒ $\tt\dfrac{\sf{x_1} + {x_2}}{2}$ = 4

⇒ $\tt\dfrac{5k + (-k)}{2}$ = 4

⇒ $\tt\dfrac{5k - k}{2}$ = 4

⇒ $\tt\dfrac{\cancel{4}k}{\cancel{2}}$ = 4

⇒ $\tt{2k = 4}$

⇒ $\tt{k = 4 - 2}$

➠ $\tt{k = 2}$

$\therefore$ The value of k is $\boxed{\bold{\large{2}}}$

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