If (a, b) is the mid - point of the line segment joining the points A(10, - 6), B(k, 4) and a – 2b = 18, then find the value of k and the distance AB.

Given,

(a, b) is the mid - point of line segment AB

Since, mid - point of a line segment having points (x_{1}, y_{1}) and (x_{2}, y_{2}) =

So,

(a, b) =

Now, equating coordinates on both sides, we get

a = …. (i)

And

b = - 1 …(ii)

Given,

a – 2b = 18

Using equation (ii);

a – 2( - 1) = 18

a + 2 = 18

a = 16

Using equation (i);

16 =

32 = 10 + k

k = 22

Hence, the required value of k is 22

→ k = 22

∴ A = (10, - 6),

B = (22, 4)

Now,

By using the distance formula;

Distance between two points (x_{1}, y_{1}) and (x_{2}, y_{2});

So,

Distance between A (10, - 6) and B (22, 4)

AB =

=

Hence, the distance of AB is

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