**Finding the Perpendicular Bisector of a Line Segment**

To find the coordinates of the point where the perpendicular bisector of the line segment joining points A (4,8) and B (6,1) intersects the y-axis, we can follow a step-by-step process.

**Step 1: Find the Midpoint of the Line Segment**

The first step is to find the midpoint of the line segment joining points A and B. The midpoint can be calculated using the formula:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

In this case, the coordinates of A are (4,8) and the coordinates of B are (6,1). Plugging these values into the midpoint formula, we get:

Midpoint = ((4 + 6) / 2, (8 + 1) / 2)
= (10 / 2, 9 / 2)
= (5, 4.5)

So, the midpoint of the line segment AB is (5, 4.5).

**Step 2: Calculate the Slope of the Line Segment**

Next, we need to calculate the slope of the line segment AB. The slope can be determined using the formula:

Slope = (y2 - y1) / (x2 - x1)

Using the coordinates of A and B, we can substitute the values into the slope formula:

Slope = (1 - 8) / (6 - 4)
= (-7) / 2
= -3.5

Therefore, the slope of the line segment AB is -3.5.

**Step 3: Determine the Negative Reciprocal of the Slope**

To find the slope of the perpendicular bisector, we need to determine the negative reciprocal of the slope of AB. The negative reciprocal is obtained by taking the negative of the inverse of the slope. In this case, the negative reciprocal of -3.5 is 1/3.5 or -1/3.5.

**Step 4: Find the Equation of the Perpendicular Bisector**

Using the midpoint and the negative reciprocal slope, we can find the equation of the perpendicular bisector. The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Since we have the midpoint (5, 4.5) and the slope -1/3.5, we can substitute these values into the equation:

4.5 = (-1/3.5)(5) + b

Simplifying the equation:

4.5 = (-5/3.5) + b
b = 4.5 + (5/3.5)
b = 4.5 + 1.4286
b = 5.9286

Therefore, the equation of the perpendicular bisector is y = (-1/3.5)x + 5.9286.

**Step 5: Find the Intersection Point with the y-Axis**

To find the point where the perpendicular bisector intersects the y-axis, we need to determine the x-coordinate when y = 0. By substituting y = 0 into the equation of the perpendicular bisector, we can solve for x.

0 = (-1/3.5)x + 5.9286

Take (x1, y1) =(4, 8) (x2, y2) =(6, 1)
Then, at y axis, therefore (x, y) =(0,y)Take xBy section formula,X=(m1x2+m2x1÷m1m2)
i.e, 0=(m1×6+m2×4÷m1+m2)
You will get the ratio.
Then find out coordinates of the point by applying full section formula.