In an optics experiment, with the position of the object fixed, a student varies the position of a convex lens and for each position, the screen is adjusted to get a clear image of the object. A graph between the object distance u and the image distance v, from the lens, is plotted using the same scale for the two axes. A straight line passing through the origin and making an angle of 45^ \circ with the x -axis meets the experimental curve at P. The coordinates of P will be :

Geometrical Optics PYQ — JEE Main 2009 · Shift 0 · Q55 with Solution

View written solutionFree

Correct answer: D

For a convex lens, the object distance $u$ and image distance $v$ satisfy

\frac{1}{f}=\frac{1}{u}+\frac{1}{v}.

So the experimental curve on the $u$ - $v$ graph is given by

\frac{1}{u}+\frac{1}{v}=\frac{1}{f}.

The line passes through the origin and makes an angle $45^\circ$ with the $x$ -axis.

Hence its slope is

\tan 45^\circ = 1,

so its equation is

v=u.

Since the same scale is used on both axes, this indeed represents all points where object distance equals image distance.

At point $P$ , we must have

u=v.

Substitute $v=u$ into the lens formula:

\frac{1}{f}=\frac{1}{u}+\frac{1}{u}=\frac{2}{u}.

Therefore,

u=2f.

And since $v=u$ ,

v=2f.

So the coordinates of $P$ are

(2f,2f).

Therefore, the correct answer is

\boxed{(2f,2f)}.