If \((a,b)\) is the midpoint of the segment joining \(A(10,-6)\) and \(B(k,4)\) and \(a-2b=18\), find \(k\) and \(|AB|\).
\(k=22\), \(|AB|=2\sqrt{61}\)
Step 1: Recall the midpoint formula
The midpoint of a line joining points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) \]
Step 2: Apply to given points
Point \(A(10, -6)\), point \(B(k, 4)\).
So, midpoint \((a,b)\) is:
\[ a = \dfrac{10 + k}{2}, \quad b = \dfrac{-6 + 4}{2} \]
Step 3: Simplify the \(y\)-coordinate of midpoint
\[ b = \dfrac{-6 + 4}{2} = \dfrac{-2}{2} = -1 \]
Step 4: Use the condition \(a - 2b = 18\)
Substitute values:
\[ a - 2b = 18 \]
\[ \dfrac{10 + k}{2} - 2(-1) = 18 \]
\[ \dfrac{10 + k}{2} + 2 = 18 \]
Step 5: Solve for \(k\)
Subtract 2 from both sides:
\[ \dfrac{10 + k}{2} = 16 \]
Multiply both sides by 2:
\[ 10 + k = 32 \]
Subtract 10:
\[ k = 22 \]
Step 6: Find length of \(|AB|\)
Formula for distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is:
\[ |AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substitute \(A(10,-6)\), \(B(22,4)\):
\[ |AB| = \sqrt{(22 - 10)^2 + (4 - (-6))^2} \]
\[ |AB| = \sqrt{(12)^2 + (10)^2} \]
\[ |AB| = \sqrt{144 + 100} = \sqrt{244} \]
\[ |AB| = 2\sqrt{61} \]