Tip: $

No need to find out both x & y co-ordinate, as soon as you find the $x co-ordinate $by mid-point theorem it comes out 0. And looking to the given option only one option satisfy this.$

Detailed Solution:$

We know that, if line segment formed by joining points A (x₁, y₁) and B (x₂, y₂) is divided by point P (x₃, y₃) to a ratio of m: n then:

$({x_3} = \frac{{m{x_2} + n{x_1}}}{{m + n}}\;\& \;{y_3} = \frac{{m{y_2} + n{y_1}}}{{m + n}})$

The center divides the diameter into two equal parts.

∴ m : n = 1 : 1

Given, co-ordinates of center is (1,2) and co-ordinates of one end of diameter is (2,4).

Let the coordinates of other end of the diameter is (a, b)

$(\Rightarrow 1 = \frac{{1 \times a + 1 \times 2}}{2}\;and\;2 = \frac{{1 \times b + 1 \times 4}}{2})$

⇒ a = 0 and b = 0