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

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

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Solution
 
mid-point\, formula\, x= \frac{x_{1}+x_{2}}{2},y= \frac{y_{1}+y_{2}}{2}
Point P (a, b) divide A(10, –6) and B(k, 4) in two equal parts.
a= \frac{10+k}{2}                b= \frac{-6+4}{2}= \frac{-2}{2}= -1
Given :  a – 2b = 18
Put b = –1
a – 2(–1) = 18
a = 18 – 12
a = 16
Now, a= \frac{10+k}{2}
16= \frac{10+k}{2}
32 = 10 + k
32 – 10 = k
22 = k
\therefore A (10, –6), B(22, 4)
AB= \sqrt{\left ( 22-10 \right )^{2}+\left ( 4+6 \right )^{2}}
= \sqrt{\left ( 12 \right )^{2}+\left ( 10 \right )^{2}}
= \sqrt{144+100}
= \sqrt{244}
=2 \sqrt{61}

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