Equation of circle in term of end points of its diameter is ( x- x1)(x-x2) + (y-y1)(y-y2) = 0 where (x1 ,y1) ,
(x2, y2) is end points. now compare the equation on putting end points with given equation of circle to get value of a.

Given:
Two points (1,2) and (2,a) are the extremities of a diameter of the circle.

To find:
The value of a.

Approach:
To find the value of 'a', we need to determine the equation of the circle using the given information.
Once we have the equation of the circle, we can substitute the x-coordinate of the second given point to find the value of 'a'.

Solution:

Step 1: Determine the equation of the circle
The equation of a circle is given by:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.

In this case, we are given two points on the diameter of the circle, which means the center of the circle lies on the perpendicular bisector of the line segment connecting these two points.

Step 2: Find the midpoint of the line segment connecting the given points
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Using the given points (1,2) and (2,a), we can find the midpoint as follows:
Midpoint = ((1 + 2)/2, (2 + a)/2) = (3/2, (2 + a)/2)

Step 3: Find the slope of the line segment connecting the given points
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
Slope = (y2 - y1) / (x2 - x1)

Using the given points (1,2) and (2,a), we can find the slope as follows:
Slope = (a - 2) / (2 - 1) = (a - 2)

Step 4: Find the negative reciprocal of the slope
The negative reciprocal of a slope is obtained by changing its sign and taking the reciprocal.
Negative Reciprocal = -1 / Slope

So, the negative reciprocal of (a - 2) is -1 / (a - 2).

Step 5: Use the midpoint and negative reciprocal of slope to determine the equation of the perpendicular bisector
Using the point-slope form of a line, the equation of the perpendicular bisector passing through the midpoint is given by:
y - y1 = m(x - x1)

Substituting the values of the midpoint (3/2, (2 + a)/2) and the negative reciprocal of slope -1 / (a - 2), we have:
y - (2 + a)/2 = -1 / (a - 2)(x - 3/2)

Simplifying the equation:
2y - 2(a + 2) = -2 / (a - 2)(2x - 3)

Multiplying throughout by (a - 2) to eliminate the denominator:
2y(a - 2) - 2(a + 2)(a - 2) = -2(2x - 3