The line 4x + ky = 20 passes through the points A (8, -4) and B (b, 2b), where k and b are constants.
- Find the values of k and b.
- Find the coordinates of the mid-point of AB.
Solution
Step-by-step Solution:
- Substitute point A (8, -4) into the line equation 4x + ky = 20:
\(4(8) + k(-4) = 20\)
\(32 - 4k = 20\)
Solve for \(k\):
\(32 - 20 = 4k\)
\(12 = 4k\)
\(k = 3\) - Substitute point B (b, 2b) into the line equation:
\(4b + 3(2b) = 20\)
\(4b + 6b = 20\)
\(10b = 20\)
Solve for \(b\):
\(b = 2\) - Find the mid-point of AB:
Mid-point formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)
\(A (8, -4), B (2, 4)\)
Mid-point: \(\left( \frac{8 + 2}{2}, \frac{-4 + 4}{2} \right) = (5, 0)\)
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