Class 10 Mathematics — Circles Notes

1. Position of a Line with Respect to a Circle

A line and a circle can have three possible relationships:

Position	Common points	Name
Line does not meet circle	0	Non-intersecting line
Line cuts circle at two points	2	Secant
Line touches circle at exactly one point	1	Tangent

A tangent touches the circle at only one point, called the point of contact.

Important idea:
A tangent can be viewed as a special limiting case of a secant in which the two intersection points merge into one.

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2. Tangent to a Circle

Definition:
A tangent is a line that intersects a circle at exactly one point.

The point where the tangent meets the circle is the point of contact.

Key facts

• There is exactly one tangent at any particular point of a circle.
• For a given secant direction, at most two parallel tangents can be drawn to a circle.

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3. Most Important Theorem

Theorem 10.1

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

If a tangent touches the circle at $P$, and $O$ is the centre, then:$$\boxed{OP\perp \text{tangent}}$$

Therefore,$$\angle OPT=90^\circ$$

Why?

For any other point $Q$ on the tangent:$$OQ>OP$$

Thus, $OP$ is the shortest distance from the centre $O$ to the tangent line. The shortest distance from a point to a line is perpendicular to the line.

Consequences

• The radius through the point of contact is perpendicular to the tangent.
• There is only one tangent at a given point.
• The line containing this radius is sometimes called the normal to the circle.

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4. Number of Tangents from a Point

The number of tangents depends on where the point lies.

Point inside the circle

$$\boxed{0\text{ tangents}}$$

Every line through an interior point cuts the circle at two points.

Point on the circle

$$\boxed{1\text{ tangent}}$$

There is exactly one tangent at that point.

Point outside the circle

$$\boxed{2\text{ tangents}}$$

Two tangents can be drawn from an external point.

Remember

$$\boxed{ \begin{array}{c|c} \text{Position of point}&\text{Number of tangents}\\ \hline \text{Inside}&0\\ \text{On circle}&1\\ \text{Outside}&2 \end{array}}$$

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5. Length of a Tangent

If $P$ is outside a circle and a tangent from $P$ touches the circle at $T$, then $PT$ is called the length of the tangent from PP.

Suppose two tangents $PQ$ and $PR$ are drawn from the same external point $P$.

Theorem 10.2

$$\boxed{PQ=PR}$$

The lengths of tangents drawn from an external point to a circle are equal.

Proof idea

Join $OP,OQ,OR$.

Because radius ⟂ tangent:$$\angle OQP=\angle ORP=90^\circ$$

Also,$$OQ=OR$$

because they are radii, and $OP$ is common.

Therefore,$$\triangle OQP\cong\triangle ORP$$

by RHS congruence.

Hence,$$\boxed{PQ=PR}$$

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6. Important Result About the Angle Between Two Tangents

If two tangents are drawn from an external point $P$, then:$$PQ=PR$$

So $\triangle PQR$ is isosceles.

Also, the line joining the external point to the centre bisects the angle between the two tangents:$$\boxed{\angle QPO=\angle OPR}$$

Thus, the centre lies on the angle bisector of the angle formed by the two tangents.

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7. Useful Results for Problems

A. Tangent + radius

If $PT$ is tangent at $T$:$$\boxed{OT\perp PT}$$

So triangle $OPT$ is right-angled at $T$.

Using Pythagoras:$$\boxed{OP^2=OT^2+PT^2}$$

Hence,$$PT=\sqrt{OP^2-OT^2}$$

This is one of the most useful formulas in numerical questions.

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B. Chord touched by a smaller concentric circle

If a chord of a larger circle touches a smaller concentric circle, the radius drawn to the point of contact is perpendicular to the chord.

Since the perpendicular from the centre to a chord bisects the chord:$$\boxed{AP=PB}$$

So the point of contact is the midpoint of the chord.

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8. High-Value Exam Relationships

For two tangents from an external point $T$:$$TP=TQ$$

and$$OP\perp TP,\qquad OQ\perp TQ$$

Also, if the angle between the tangents is $\angle PTQ$, then:$$\boxed{\angle PTQ=2\angle OPQ}$$

as established in the chapter’s Example 2.

Another important relationship is:$$\boxed{\angle(\text{two tangents})+\angle POQ=180^\circ}$$

where $P,Q$ are the points of contact and $O$ is the centre. This is directly reflected in the chapter’s exercises.

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9. Problem-Solving Strategy

When you see a tangent, immediately write:$$\boxed{\text{Radius at contact}\perp\text{tangent}}$$

When you see two tangents from the same external point, immediately write:$$\boxed{\text{Tangent}_1=\text{Tangent}_2}$$

Then look for:

• RHS congruence
• Pythagoras theorem
• Isosceles triangle
• Angle bisector
• Perpendicular from centre to chord
• Similar triangles

These are the main tools repeatedly used in the chapter’s proofs and examples.

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10. One-Minute Revision Sheet

Definitions

• Secant: line cutting a circle at two points.
• Tangent: line touching a circle at one point.
• Point of contact: point where tangent touches the circle.
• Normal: line through the centre and point of contact.

Must-remember theorems

$$\boxed{\text{Tangent}\perp\text{radius at point of contact}}$$$$\boxed{\text{Tangents from an external point are equal}}$$

Number of tangents

$$\boxed{\text{Inside}\to0,\quad \text{On circle}\to1,\quad \text{Outside}\to2}$$

Numerical formula

$$\boxed{PT^2=OP^2-OT^2}$$

Angle result

$$\boxed{\angle PTQ=2\angle OPQ}$$