The intensity of the image produced by a telescope is directly related to the area of the objective lens or mirror. So, if the diameter of the objective is doubled, the area of the objective increases by a factor of four.

Let's understand this concept in more detail:

1. Relationship between intensity and area:
- The intensity of an image formed by a telescope is directly proportional to the area of the objective lens or mirror.
- This means that if the area of the objective is increased, the intensity of the image will also increase.

2. Relationship between diameter and area:
- The area of a circle is directly proportional to the square of its diameter.
- Mathematically, the area of a circle is given by the formula A = πr^2, where r is the radius of the circle.
- Since the diameter of a circle is twice the radius, we can rewrite the formula as A = π(d/2)^2, where d is the diameter of the circle.
- Simplifying the equation, we get A = (π/4)d^2.
- This equation shows that the area of a circle is proportional to the square of its diameter.

3. Doubling the diameter:
- If we double the diameter of the objective lens or mirror, the new diameter becomes 2d.
- Using the equation for area, the new area becomes (π/4)(2d)^2 = (π/4)4d^2 = πd^2.
- Comparing the new area (πd^2) with the original area (A = (π/4)d^2), we can see that the new area is four times the original area.
- Since intensity is directly proportional to the area, the intensity of the image will also increase by a factor of four.

Therefore, if the diameter of the objective of a telescope is doubled, the intensity of the image is increased by a factor of four. This is because the area of the objective increases by a factor of four when the diameter is doubled.