If the coordinates of the mid-points of the line joining the points (3a, 4) and ( - 2, 2b) are (5, a), then

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 $(x_1, y_1)$ and $(x_2, y_2)$, 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 $\frac{3a - 2}{2}$.

Step 4: Solve for 'a'

We solve the equation $5 = \frac{3a - 2}{2}$ 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 $\frac{4 + 2b}{2}$.

Step 6: Substitute 'a' and solve for 'b'

Since we found $a = 4$, we substitute this value into the equation $a = \frac{4 + 2b}{2}$. This gives $4 = \frac{4 + 2b}{2}$. 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 $b = 2$.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

All Coordinate Geometry solutions →