Draw two circles touch each other externally in a common point they have one common tangent

We will discuss here about common tangents to two circles.

1. No Common Tangent:

If one circle line completely inside another circle without cutting or touching it at any point then the circles will have no common tangent.

2. One Common Tangent:

If two circles touch each other internally at one point, they will have one common tangent.

3. Two Common Tangents:

If two circles intersect each other at two points, they will have two common tangents.

4. Three Common Tangents:

If two circles touch each other externally at one point, they will have three common tangents.

5. Four Common Tangents:

If two circles do not touch or intersect each other and one does not lie inside the other, they will have four common tangents.

AB and CD are called direct common tangents, and MN and XY are called transverse common tangents.

10th Grade Math

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Here we will prove that if two circles touch each other, the point of contact lies on the straight line joining their centres.

Case 1: When the two circles touch each other externally.

Given: Two circles with centres O and P touch each other externally at T.

To prove: T lies on the line OP.

Construction: Draw a common tangent XY through the point of contact T. Join T to O and P.

Proof:

Case 2: When the two circles touch each other internally at T.

To prove: T lies on OP produced.

Construction: Draw a common tangent XY through the point of contact T. Join T to O and P.

Proof:

Note: Let two circles with centres O and P touch each other at T. Let OT = r1 and PT = r2 and r1 > r2.

Let the distance between their centres = OP = d.

It is clear from the figures that

• When the circles touch externally, d = r1 + r2.

• When the circles touch internally, d = r1 - r2.

10th Grade Math

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Answer

In the above figure, we can see that one tangent is passing from the point of contact of both the circles.

The second and third tangent is passing from the top and bottom point of both the circles. Therefore, from two circles touching externally, three tangents can pass. So, the correct answer is three.Additional Information: Tangent is a straight line which touches the circle at a particular point. The line from the centre of the circle and the tangent forms at 90 degree angle. when considered two circles. There can be three tangents in common. The one tangent will be at the point of touching where the two circles are touching each other. The point of contact of both the circles.The second tangent will be at the bottom which will touch both the circles. The common tangent which is formed by touching the bottom ends of both the circles. The third tangent will be at the top which will also pass by touching both the circles. The common tangent which is formed by touching the top ends of both the circles.

Therefore, The common tangents which can be drawn if two circles are touching externally at a single point = 3.

Note: Many a times, most of the students forget to take the two tangents which are touching the two circles from the top and the bottoms respectively. So, do consider that.

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