Question

If mid point of the line segment joining (2a,4) and (−2,3b) is (1,2a+1), then the values of a and b are given by

A. a=2,b=-2
B. a=b=2
C. a=1=b
D. a=-2,b=2

Answer 1

To find the values of a and b, we use the midpoint formula which results in a system of equations. Solving these equations, we get a = 2 and b = 2. Hence, the correct answer is option B: a=b=2. Therefore, the values of a and b are both 2, which matches option B.

Step-by-step explanation:

The question asks us to find the values of a and b given that the midpoint of the segment connecting the points (2a, 4) and (-2, 3b) is (1, 2a+1). We will use the formula for calculating the midpoint of a line segment, which is the average of the x-coordinates and y-coordinates of the endpoints respectively. So, the midpoint M(x, y) would have the coordinates M((x1 + x2)/2, (y1 + y2)/2).

The midpoint of a line segment is the average of the coordinates of its endpoints. In this case, we are given that the midpoint of the line segment joining (2a,4) and (-2,3b) is (1,2a+1). So, we can set up the following equations:

(2a + -2)/2 = 1 and (4 + 3b)/2 = 2a + 1

Simplifying the equations, we get: a - 1 = 1 and 2a + b = 3

Solving these two equations simultaneously, we find that a = 1 and b = 2.

Substituting the given points and midpoint into the formula, we get the equations:

1. (2a - 2) / 2 = 1
2. (4 + 3b) / 2 = 2a + 1

Solving these equations:

• 2a - 2 = 2 → 2a = 4 → a = 2
• 4 + 3b = 2(2a + 1) → 4 + 3b = 2(2(2) + 1) → 4 + 3b = 10 → 3b = 6 → b = 2

Therefore, the values of a and b are both 2, which matches option B.