Normal: Shortest Distance (Part 2)

Shortest distance between a line and a circle

Now we have some ideas about shortest distances, I’ll now move on to finding out the shortest distance between a line and a circle.

Here comes another figure.

What will be the shortest distance between the given line and the circle?

We’ll use our knowledge from the previous lesson to find the shortest distance.

In the figure above, the shortest distance between S and the line is SR (i.e. the perpendicular distance). And the shortest distance between P and the circle is PQ (i.e. along the line joining the centre).

But shortest distance between R and the circle is not RS. Similarly, the shortest distance between Q and the line will not be PQ. That means, we can have shorter distances than both RS and PQ.

What if we could have both? That is, a line passing through the centre of the circle as well as perpendicular to the given line? Have a look.

This in fact will be the shortest distance between the line and the circle. This is because, it is both the shortest distance between Q and the line, and the shortest distance between P and the circle.

Not convinced? I’ll give you another logic.

The distance of the point P from the line will be the distance of the centre of the circle from the line minus the radius. That is PQ = OP – r.

Now we have to minimize PQ. We cannot do anything to r, which is constant. That means, to minimize PQ, we have to minimize OP, which is the distance of the point O from the line.

And we know how to do that! By dropping a perpendicular from O on the line. Have a look again.

And this minimum distance can be calculated as distance of the point O from the line, minus the radius. That is, the shortest distance will be OP – r, where P is the foot of perpendicular from the centre O to the line.

I hope you remember how to calculate OP. Head back here, in case you forgot.

Finally, we can discuss the shortest distance between two circles.

Shortest distance between two circles

I’ll skip the explanation here, and leave it up to you to understand, that he shortest distance will be along the line joining the centres of the two circles. And this can be calculated as C₁C₂ – r₁ – r₂

Lesson Summary

1. The shortest distance between a circle and a line (which do not intersect) is equal to OP – r, where OP is the perpendicular distance of the centre O of the circle from the line, and r is the radius of the circle.
2. The shortest distance between two circles is given by C₁C₂ – r₁ – r₂, where C₁C₂ is the distance between the centres of the circles and r­₁ and r­₂ are their radii. Note that this expression is valid only when the two circles do not intersect, and both lie outside each other.

So, what happens when one circle lies inside the other?

I’ll discuss this case during examples. But try this case on your own first.

And why is this discussion happening under the topic normal? Can you see the normals now? Notice that for both the cases above, shortest distance was found along the line which is perpendicular to both the curves, i.e. the common normal to the two curves. And that will almost always happen.

The shortest distance (and also the longest, in some cases) between two curves almost always occurs along the common normal to the two curves. I’ve given you an intuitive idea (I hope), but there are ways to prove it. Don’t worry about those proofs now.

See you in the next lesson then, where I’ll discuss related examples.