Option 1 is the correct answer. In a long-sighted eye, the eyeball is shorter than normal, causing light rays to converge behind the retina instead of directly onto it. As a result, the person with long-sightedness can see distant objects clearly, but has difficulty focusing on close objects.

To correct this, a convex lens is used. A convex lens is thicker in the middle and thinner at the edges. When placed in front of the long-sighted eye, it helps to refract (bend) the light rays entering the eye, so that they converge on the retina. This helps to bring the image into focus on the retina, allowing the person to see clearly at both near and far distances.

Option 2 is incorrect because a concave lens is used to correct short-sightedness (myopia), where distant objects appear blurry. A concave lens is thinner in the middle and thicker at the edges. It helps to diverge the incoming light rays, so that they converge properly onto the retina in a myopic eye.

Options 3 and 4 are also incorrect as they do not address the primary issue of long-sightedness, which is difficulty in focusing on near objects. A convex lens is needed to correct this problem.