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Question
If (a,b) is the mid-point of the line segment joining the points A (10, - 6) , B (k,4) and a - 2b = 18 , find the value of k and the distance AB.
Solution
It is given that A(10, −6) and B(k, 4).
Suppose (a, b) be midpoint of AB. Then,
\[a = \frac{10 + k}{2}, b = \frac{- 6 + 4}{2} = \frac{- 2}{2} = - 1\]
\[\text{ Now } , a - 2b = 18\]
\[ \Rightarrow a = 18 - 2 = 16\]
\[\text{ Therefore }, \]
\[16 \times 2 = 10 + k\]
\[ \Rightarrow k = 22\]
\[\text{ Further } , \]
\[AB = \sqrt{\left( 22 - 10 \right)^2 + \left( 4 + 6 \right)^2}\]
\[ = \sqrt{144 + 100}\]
\[ = 2\sqrt{61}\]
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