Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): The shapes of areas shown in the Zenithal Stereographic (Polar Case) Projection are maintained correctly.
Reason (R): In the Zenithal Stereographic (Polar Case) Projection the scales along the meridians increase away from the centre at the same rate at which they increase along the parallels.
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

A projection is conformal if the scale at any point is independent of direction. Stereographic is the only azimuthal projection that is conformal.

Answer 1

The Correct Option is A

Concept: The Zenithal Stereographic Projection is a "conformal" (orthomorphic) projection. Conformal means that the projection preserves the correct shapes of small areas and maintains correct angles around any point.

Step 1: Analyzing the property of Conformality (A).
In the Polar Case of this projection, the point of tangency is one of the poles. A key characteristic of this projection is that it is conformal. This means that while the size of areas might be distorted as we move away from the pole, the local *shape* of any small area remains true to its appearance on the globe. Thus, Assertion (A) is correct.

Step 2: Analyzing the Scale Relationship (R).
For a map to be conformal, the scale at any point must be the same in all directions. In the Zenithal Stereographic projection, as you move away from the center (the pole), the scale along the meridians (radial lines) increases. Crucially, the scale along the parallels (circles) also increases. The math of the stereographic projection ensures that these two scales increase at exactly the same rate.

Step 3: Determining the Explanation.
Because the meridian scale and the parallel scale increase at the same rate, the ratio between vertical and horizontal dimensions is preserved ($1:1$). This balanced scale increase is exactly what allows the shapes of areas to be maintained correctly. Therefore, Reason (R) is the mathematical explanation for Assertion (A).