A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length f, as shown in the figure. Which of the figures shown in the four options qualitatively represent(s) the shape of the image of the bent wire? (These figures are not to scale.)

Geometrical Optics PYQ — JEE Advanced 2018 · Shift 2 · Q43 with Solution

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Correct answer: D

Key idea: image of each point of the wire in a concave mirror

For a concave mirror, if an object point is at distance $u$ from the pole, its image distance $v$ is given by

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

So different points of the bent wire generally form images at different distances from the mirror.

Also, the wire is a right-angled triangle, so its three sides must be considered separately:

one side along the principal axis,
one side perpendicular to the principal axis,
one slanted side.

The qualitative image shape is obtained by seeing how each side transforms.

Image of the side perpendicular to the principal axis

Suppose this vertical side is at a fixed object distance $u$ . Since all its points have the same $u$ , all image points have the same image distance $v$ .

Hence, this side remains a straight line perpendicular to the principal axis in the image.

So the corresponding side of the image should be a vertical straight segment.

Image of the side lying along the principal axis

For the side along the principal axis, different points have different object distances $u$ . Thus their image distances $v$ change according to

v=\frac{fu}{u-f}.

Therefore, the image of this horizontal side does not remain a straight horizontal segment of the same kind as the object; rather, its points lie along the principal axis itself, but at non-uniformly related positions.

So this side still appears along the axis, joining the images of the appropriate endpoints.

Image of the slanted side

This is the most important part.

Take a point $P(x,y)$ on the slanted side. Its image coordinates depend on its distance from the mirror. Since points on the slanted side have varying $u$ , both:

the image distance along the axis, and
the transverse magnification

change from point to point.

Magnification is

m=-\frac{v}{u}.

Since $u$ changes along the slanted side, $m$ also changes.

Therefore, the image of the slanted straight side is generally not a straight line. It becomes a curved line.

So the correct figure must show:

one vertical straight side,
one axial side,
and the hypotenuse becoming a curve.

Nature of the curvature

For a concave mirror, as object distance changes, image distance and magnification vary nonlinearly. Hence a straight oblique segment bends into a curve connecting the images of its endpoints.

Thus the correct option is the one where the triangular wire’s image is distorted so that the oblique side is curved, not straight.

Matching with the given options

Among the four figures, the one that qualitatively represents this behavior is Option D.

Final answer

Therefore, the correct choice is

\boxed{D}

Comparison with stored correct answer

Stored correct answer: $D$

My derived answer is also $D$ , so they agree.

More from Geometrical Optics

View written solutionFree

Correct answer: D

Key idea: image of each point of the wire in a concave mirror

For a concave mirror, if an object point is at distance $u$ from the pole, its image distance $v$ is given by

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

So different points of the bent wire generally form images at different distances from the mirror.

Also, the wire is a right-angled triangle, so its three sides must be considered separately:

one side along the principal axis,
one side perpendicular to the principal axis,
one slanted side.

The qualitative image shape is obtained by seeing how each side transforms.

Image of the side perpendicular to the principal axis

Suppose this vertical side is at a fixed object distance $u$ . Since all its points have the same $u$ , all image points have the same image distance $v$ .

Hence, this side remains a straight line perpendicular to the principal axis in the image.

So the corresponding side of the image should be a vertical straight segment.

Image of the side lying along the principal axis

For the side along the principal axis, different points have different object distances $u$ . Thus their image distances $v$ change according to

v=\frac{fu}{u-f}.

So this side still appears along the axis, joining the images of the appropriate endpoints.

Image of the slanted side

This is the most important part.

Take a point $P(x,y)$ on the slanted side. Its image coordinates depend on its distance from the mirror. Since points on the slanted side have varying $u$ , both:

the image distance along the axis, and
the transverse magnification

change from point to point.

Magnification is

m=-\frac{v}{u}.

Since $u$ changes along the slanted side, $m$ also changes.

Therefore, the image of the slanted straight side is generally not a straight line. It becomes a curved line.

So the correct figure must show:

one vertical straight side,
one axial side,
and the hypotenuse becoming a curve.

Nature of the curvature

For a concave mirror, as object distance changes, image distance and magnification vary nonlinearly. Hence a straight oblique segment bends into a curve connecting the images of its endpoints.

Thus the correct option is the one where the triangular wire’s image is distorted so that the oblique side is curved, not straight.

Matching with the given options

Among the four figures, the one that qualitatively represents this behavior is Option D.

Final answer

Therefore, the correct choice is

\boxed{D}

Comparison with stored correct answer

Stored correct answer: $D$

My derived answer is also $D$ , so they agree.

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Geometrical Optics question

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