### Video Transcript

The diagram shows a thin concave
lens. The lens is symmetrical. The point marked 𝑃 is one of the
foci of the lens. Using the grid, determine which of
the points — I, II, III, or IV — is the other focus of the lens.

In this question, we are shown a
diagram with a concave lens on a grid, with one of the lens’s foci labeled 𝑃 and
four other points. We are asked to figure out which of
the four points is the other focus of this concave lens.

Now, remember that the shape of a
concave lens can be created by overlapping a cylinder and two spheres, like
shown. If the overlapped material from the
cylinder is removed, we are left with the shape of a concave lens. If we consider a two-dimensional
cross section of a concave lens, it looks like the shape made by removing two equal
parts of a circle from opposite sides of a rectangle. The shape that is produced is
thicker along the edges and thinnest in the middle. This shape causes concave lenses to
act as diverging lenses. This means that light rays
traveling through a concave lens will change directions to diverge or spread out
from each other.

The centers of the circles that we
saw to determine the shape of the cross section of a concave lens are the centers of
curvature of the lens. If we connect the centers of
curvature with a line, the line passes through the center of the lens. This line is called the optical
axis. If a ray of light traveled along
this line through the lens, it would not change its direction as it passed through
the lens.

Let’s see what happens when a few
rays of light traveling parallel to the optical axis enter the lens. These outer rays change
direction. The rays will appear to diverge
from a point on the side of the lens from which they enter the lens. If we trace the path of these rays
after they exit the lens back through the lens, we find that the extended paths do
all converge at the same point. This point is called the focal
point of the lens. A concave lens is symmetrical, so
it has two focal points, one on either side of the lens, each equal distances from
the center of the lens.

So now we know how to find the
focal point of a concave lens. Let’s go back to the diagram given
in the question and use our point 𝑃 to draw some rays. We can pick two points along the
vertical axis of the lens and draw lines from the focal point 𝑃 through these
points. Note that these aren’t light rays
we’re drawing, but at this point just dashed lines. Let us suppose that two light rays
were originally parallel to the optical axis of the lens when they entered the lens
at these two points we’ve picked. The lines on the opposite side to
point 𝑃 are showing us the paths that these light rays would travel after entering
the lens. We can measure the horizontal
distance from point 𝑃 to the center of the lens, which happens to be three squares
on this grid.

Because concave lenses are
symmetrical lenses, the focal length for the two foci of the lens will be the
same. So we can measure three squares
from the center of the lens to the right. We can see that point II is three
squares away from the center of the lens. So this must be our second focal
point. Let’s draw lines from point II
through the points we made on the vertical axis of the lens. Because this lens is symmetrical,
these lines should be reversed copies of the lines from point 𝑃. And this is exactly what we see
when we draw them. Therefore, point II is the second
focal point of the lens. Point II is the correct answer.