Tangent to a Circle

We know that circles and lines are two distinct shapes that have very little in common. But what happens when the two of them meet or intersect at any single point? This gives rise to a tangent. A tangent to a circle is a line that touches the circle at a single point. The point to tangency is where the circle meets the point. A tangent is also perpendicular to the radius of the circle by which it intersects. It can be considered for any curved shape. A tangent is a line has its equation. In this chapter, we will learn tangent to a circle in various other forms. We will also see the equation of tangent to a circle and tangent to a circle formula.

Definition of Tangent to Circle

A line that joins two close points from a point on the circle is known as a tangent. In simple words, we can say that the lines that intersect the circle exactly in one single point are tangents. Only one tangent can be at a point to circle. The point where a tangent touches the circle is known as the point of tangency. The point where the circle and the line intersect is perpendicular to the radius. As it plays a vital role in the geometrical construction there are many theorems related to it which we will discuss further in this chapter.

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Here, point O is the radius, point P is the point of tangency.

Various Conditions of Tangency

Only when a line touches the curve at a single point it is considered a tangent. Or else it is considered only to be a line. Hence, we can define tangent based on the point of tangency and its position with respect to the circle. 

1. When point lies on the circle

2. When point lies inside the circle

3. When point lies outside the circle

When Point Lies on the Circle

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Here, from the figure, it is stated that there is only one tangent to a circle through a point that lies on the circle.

When Point Lies Inside the Circle

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In the figure above, the point P is inside the circle. Now, all the lines passing through point P are intersecting the circle at two points. therefore, no tangent can be drawn to the circle that passes through a point lying inside the circle.

When Point Lies Outside the Circle

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The above figure concludes that from a point P that lies outside the circle, there are two tangents to a circle.

Properties of Tangent

Always remember the below points about the properties of a tangent

1. A line of tangent never crosses the circle or enters it; it only touches the circle.

2. The point at which the lien and circle intersect is perpendicular to the radius

3. The tangent segment to a circle is equal from the same external point.

4. A tangent and a chord forms an angle, the angle is exactly similar to the tangent inscribed on the opposite side of the chord.

Equation of Tangent to a Circle

Below is the equation of tangent to a circle

1. Tangent to a circle equation x2+ y2=a2 at (a cos θ, a sin θ) is x cos θ+y sin θ= a

2. Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2

3. Tangent to a circle equation x2+ y2=a2 for a line y = mx +c is y = mx ± a √[1+ m2]

4. Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2

Tangent to a Circle Formula

To understand the formula of the tangent look at the diagram given below.

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Here, we have a circle with P as its exterior point. From the exterior point P the circle has a tangent at Point Q and S. A straight line that cuts the curve in two or more parts is known as a secant. So, here the secant is PR and at point Q, R intersects the circle as shown in the diagram above. So, now we get the formula for tangent-secant 

PR/PS = PS/ PQ

PS² = PQ.PR

Theorems of Tangents to Circle

Theorem 1

A radius is gained by joining the centre and the point of tangency. A tangent at the common point on the circle is at a right angle to the radius. The below diagram will explain the same where AB \[\perp\] OP

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Theorem 2

From one external point only two tangents are drawn to a circle that have equal tangent segments. A tangent segment is the line joining to the external point and the point of tangency. According to the below diagram AC = BC

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Examples of a Tangent to a Circle Formula

Example 1

In the below circle point O is the radius, PT is a tangent and OP is the radius, If PT is a tangent, then OP is perpendicular to PT.

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If OP = 3 Units and PT = 4 Units. Find the length of OT

Solution: as the radius is perpendicular to the tangent at the point of tangency, OP \[\perp\] PT

Therefore, ∠P is the right angle in the triangle OPT and triangle OPT is a right angle triangle.

Now, according to the Pythagoras theorem, we find OT.

(OP)² + (PT)² = (OT)²

3² + 4² = (OT)²

9 + 16 = (OT)²

25 = (OT)²

5 = OT

As the length cannot be negative, the length of OT is 5 units.

Example 2 

In the below diagram PA and PB are tangents to the circle. Find the value of 

1. ∠OAP

2. ∠AOB

3. ∠OBA

4. ∠ASB

5. The length of OP, PB = 7 cm (given)

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Solution: 

1. ∠OAP = 90° (Tangent is perpendicular to the radius)

2. ∠AOB + ∠APB = 180°

∠AOB + 48° = 180°

∠AOB = 180° - 48° = 132°

3. ∠OBA + ∠OAB + ∠AOB = 180° (angle sum of triangle)

2 x ∠OBA + ∠AOB = 180° (∠OBA = ∠OAB)

2 x ∠OBA + 132° = 180° (∠AOB = 132°)

∠OBA = 24°

4. ∠AOB = 2 x ∠ASB (angle at centre = 2 angle at circle)

∠ASB = ∠AOB / 2 

∠ASB = 132° / 2 = 66°

5. Cos 24° = \[\frac{7}{OP}\] ⇒ OP =  \[\frac{7}{cos24^{0}}\]