(a, 2b) is the mid-point of the line segment joining the points (10, -6) and (k, 4). If a – 2b = 7, then what is the value of k?

Concept:

Let A(x₁, y₁) and B(x₂, y₂) be any two points on the X-Y plane. Suppose point C is the mid-point of the line segment AB, then the coordinates of point C is:

 \(\left( {\frac{{{{\rm{x}}_1} + {{\rm{x}}_2}}}{2},\frac{{{{\rm{y}}_1} + {{\rm{y}}_2}}}{2}} \right)\).

Calculation:

Mid-point of (10, -6) and (k, 4) :

\(\left( {{\rm{a}},2{\rm{b}}} \right) = \left( {\frac{{10 + {\rm{k}}}}{2},\frac{{ - 6 + 4}}{2}} \right) = \left( {5 + \frac{{\rm{k}}}{2},-1} \right)\)

Therefore,

\({\rm{a}} = \frac{{\rm{k}}}{2} + 5\)

2b = - 1

Given, a – 2b = 7

\(\left( {\frac{{\rm{k}}}{2} + 5{\rm{\;}}} \right) + 1 = 7\)

\(\frac{{\rm{k}}}{2} + 6 = 7\)

\(\frac{{\rm{k}}}{2} = 1\)

k = 2