Matching Geometric Curves with Common Tangents and Normals

Geometric Analysis of Configurations:

1. (A) Two intersecting circles:

• They have two direct common tangents, so they have a common tangent ($p$).
• Any normal to a circle passes through its center. The straight line joining the centers of the two circles is a normal to both circles simultaneously. Hence, they have a common normal ($q$).
• Matching: $(\text{A}) \rightarrow (\text{p}, \text{q})$.

1. (B) Two mutually external circles:

• They have four common tangents (two direct, two transverse), so they have a common tangent ($p$).
• The line of centers is a normal to both circles, so they have a common normal ($q$).
• Matching: $(\text{B}) \rightarrow (\text{p}, \text{q})$.

1. (C) Two circles, one strictly inside the other:

• One circle is entirely contained within the interior of the other without touching, so they do not have a common tangent ($r$).
• The line connecting their centers passes through the center of both circles and is perpendicular to both circumferences, so they have a common normal ($q$).
• Matching: $(\text{C}) \rightarrow (\text{q}, \text{r})$.

1. (D) Two branches of a hyperbola:

• No tangent to one branch can touch the other branch (hyperbolic tangents lie strictly on their respective branches), so they do not have a common tangent ($r$).
• The transverse axis is the axis of symmetry and intersects both vertices perpendicularly, serving as a shared normal to both branches. Hence, they have a common normal ($q$).
• Matching: $(\text{D}) \rightarrow (\text{q}, \text{r})$.

Matching:
$(\text{A}) \rightarrow (\text{p}, \text{q}),\; (\text{B}) \rightarrow (\text{p}, \text{q}),\; (\text{C}) \rightarrow (\text{q}, \text{r}),\; (\text{D}) \rightarrow (\text{q}, \text{r})$

Correct Answer: D