If the coordinates of the mid-points of the line joining the points (3a, 4) and ( - 2, 2b) are (5, a), then
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Step-by-Step Solution
Step 1: Recall the Midpoint Formula
The midpoint formula is used to find the coordinates of the point that lies exactly halfway between two given points. If the two points are (x1,y1) and (x2,y2), then the coordinates of their midpoint (x, y) are found by averaging their respective x -coordinates and y -coordinates.
Step 2: Apply the Midpoint Formula to the given points
We are given two points, (3a,4) and (−2,2b), and their midpoint is (5,a). We substitute these values into the midpoint formula. The x -coordinate of the midpoint is 5, and the y -coordinate is a.
Step 3: Equate the x-coordinates
To find the value of a, we equate the x -coordinate of the given midpoint, which is 5, with the x -coordinate derived from the midpoint formula, which is 23a−2.
Step 4: Solve for 'a'
We solve the equation 5=23a−2 for a. First, multiply both sides by 2 to get 10=3a−2. Then, add 2 to both sides to get 12=3a. Finally, divide by 3 to find a=4.
Step 5: Equate the y-coordinates
Next, we equate the y -coordinate of the given midpoint, which is a, with the y -coordinate derived from the midpoint formula, which is 24+2b.
Step 6: Substitute 'a' and solve for 'b'
Since we found a=4, we substitute this value into the equation a=24+2b. This gives 4=24+2b. Multiply both sides by 2 to get 8=4+2b. Subtract 4 from both sides to get 4=2b. Finally, divide by 2 to find b=2.