What Is The Tangent To A Circle Definition Formula And Properties
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.
When point lies on the circle
When point lies inside the circle
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
A line of tangent never crosses the circle or enters it; it only touches the circle.
The point at which the lien and circle intersect is perpendicular to the radius
The tangent segment to a circle is equal from the same external point.
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
Tangent to a circle equation x2+ y2=a2 at (a cos θ, a sin θ) is x cos θ+y sin θ= a
Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2
Tangent to a circle equation x2+ y2=a2 for a line y = mx +c is y = mx ± a √[1+ m2]
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
∠OAP
∠AOB
∠OBA
∠ASB
The length of OP, PB = 7 cm (given)
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Solution:
∠OAP = 90° (Tangent is perpendicular to the radius)
∠AOB + ∠APB = 180°
∠AOB + 48° = 180°
∠AOB = 180° - 48° = 132°
∠OBA + ∠OAB + ∠AOB = 180° (angle sum of triangle)
2 x ∠OBA + ∠AOB = 180° (∠OBA = ∠OAB)
2 x ∠OBA + 132° = 180° (∠AOB = 132°)
∠OBA = 24°
∠AOB = 2 x ∠ASB (angle at centre = 2 angle at circle)
∠ASB = ∠AOB / 2
∠ASB = 132° / 2 = 66°
Cos 24° = \[\frac{7}{OP}\] ⇒ OP = \[\frac{7}{cos24^{0}}\]
FAQs on Tangent To A Circle Explained With Key Concepts
1. What is a tangent to a circle?
A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact. Unlike a secant, it does not cut through the circle. At the point of contact, the tangent just “grazes” the circle without entering its interior. This concept is fundamental in circle geometry and coordinate geometry.
2. What is the point of contact in a tangent?
The point of contact is the single point where the tangent line touches the circle. At this point, the radius drawn from the center of the circle to the point of contact is perpendicular to the tangent. This unique touching point distinguishes a tangent from other lines like secants or chords.
3. Why is the tangent perpendicular to the radius at the point of contact?
A tangent is perpendicular to the radius at the point of contact because the shortest distance from the center of a circle to a line is along a perpendicular. Since the radius meets the tangent at exactly one point, it forms a 90° angle with the tangent. This is a key theorem in circle geometry.
4. What is the formula of a tangent to a circle in coordinate geometry?
The equation of a tangent to a circle depends on the circle’s equation and the point of contact. For a circle x² + y² = r², the tangent at point (x₁, y₁) on the circle is xx₁ + yy₁ = r². For a general circle x² + y² + 2gx + 2fy + c = 0, the tangent at (x₁, y₁) is:
xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0. These formulas are widely used in coordinate geometry problems.
5. How do you find the length of a tangent from an external point?
The length of a tangent from an external point is found using the formula √(OP² − r²), where OP is the distance from the center to the external point and r is the radius. Steps:
- Find OP using the distance formula.
- Square OP and subtract r².
- Take the square root.
6. Are tangents drawn from an external point equal?
Yes, tangents drawn from the same external point to a circle are equal in length. If PA and PB are tangents from point P, then PA = PB. This is called the tangent theorem and is frequently used in geometry proofs and problem-solving.
7. What is the difference between a tangent and a secant?
The main difference is that a tangent touches the circle at exactly one point, while a secant intersects the circle at two points. A tangent forms a right angle with the radius at the point of contact, whereas a secant passes through the circle, creating a chord between its intersection points.
8. How do you prove that the tangent at any point of a circle is perpendicular to the radius?
To prove this theorem, show that the radius drawn to the point of contact forms a 90° angle with the tangent. Steps:
- Let O be the center and P the point of contact.
- Draw radius OP.
- Assume the tangent is not perpendicular and draw another line closer to O.
- This would intersect the circle at another point, contradicting the definition of a tangent.
9. What is the condition for a line to be tangent to a circle?
A line is tangent to a circle if the perpendicular distance from the center to the line equals the radius. In coordinate geometry, substitute the line into the circle’s equation; if the resulting quadratic has exactly one solution (discriminant D = 0), the line is a tangent. This condition ensures the line touches the circle at exactly one point.
10. Can you give an example of finding the equation of a tangent to a circle?
Yes, to find the equation of a tangent, use the standard tangent formula for the circle. Example: For the circle x² + y² = 25, find the tangent at point (3, 4). Since 3² + 4² = 25, the point lies on the circle. Using the formula xx₁ + yy₁ = r², substitute x₁ = 3, y₁ = 4, r² = 25:
3x + 4y = 25. This is the required equation of the tangent.
