Animated Solution for Mathematics - Conic Sections: Match the statements in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the $4\times 4$ matrix given in the ORS.

Before we dive into the specific cases, let us define our tools. A normal to a curve at a point is a line perpendicular to the tangent at that point.

For a circle, this is beautifully simple: every line passing through the center is a normal. For a hyperbola, the transverse axis is the line of symmetry that pierces the vertices at a perfect $90^{\circ }$ angle.

When we ask if two curves have a 'common normal,' we are asking: Is there a single line that acts as a normal to both curves?

The Circles

Imagine two circles. Whether they are intersecting or mutually external, they share a profound connection: their centers.

If you draw a line passing through the center of Circle 1 and the center of Circle 2, you have created a line that is a normal to Circle 1 and a normal to Circle 2. Thus, for any two circles, the line joining their centers is always a common normal.

This is why both intersecting and external circles satisfy the property of having a common normal. For intersecting circles, we can visualize two direct common tangents. For external circles, we have two direct tangents and two transverse tangents.

The Nested Trap

This is where many students stumble. We have a small circle inside a large one.

You might think, 'They don't touch, so they can't have a common normal.' But remember our definition: the line connecting their centers still exists and passes through both centers. Therefore, it is still a common normal.

However, the tangent situation is different. If you try to draw a line tangent to the inner circle, it will inevitably slice through the outer circle like a knife. Thus, nested circles have a common normal but absolutely no common tangent.

The Hyperbola's Symmetry

Finally, we look at the two branches of a hyperbola. These are two curves curving away from each other, separated by the vast emptiness of the asymptotes.

A common tangent is impossible here because the curves are 'back-to-back.' But look at the transverse axis—the line that connects the two vertices.

At the vertex of the left branch, this axis is perpendicular to the curve. At the vertex of the right branch, it is also perpendicular. Because this single line is a normal to both branches, the two branches of a hyperbola share a common normal.

By looking at these shapes, we have moved beyond rote memorization. We have seen that:

1. Intersecting Circles: Have common tangents and a common normal. 2. External Circles: Have common tangents and a common normal. 3. Nested Circles: Have no common tangent, but do have a common normal. 4. Hyperbola Branches: Have no common tangent, but do have a common normal.

Mathematics is not just about the equations; it is about the elegance of the visualization. When you see a problem like this, do not rush to the algebra. Close your eyes, visualize the curves, and let the geometry speak to you.