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Tangent to a Circle at a Point

In general, the word “tangent” simply means “ to touch ”. For example, when a bicycle moves along a road horizontally then the road becomes the tangent at each point when wheels roll on it.


But, how it is expressed in the form of an equation and what are its properties, are a few things that we will be dealing with in this article.

Table of Contents

  • Tangent to a Circle
  • Equation of Tangent to a Circle in Point Form
  • Equation of Tangent to a Circle in Parametric Form
  • Equation of a Tangent to a Circle in Slope Form
  • Practice Problems on Tangent to a Circle at a Point
  • FAQ’s on Tangent to a Circle at a Point

Tangent to a Circle

Tangent is basically a straight line which touches the circle at exactly one point.

For obtaining a tangent, let the secant line intersect a circle at points and . If we rotate about till coincides with , then the secant becomes the tangent at as shown in the diagram below and point is known as the point of contact.Thus a tangent can be termed as a “limiting case of secant”.


Equation of Tangent to a Circle in Point Form

The equation of  tangent to the circle at a point on the circle is where                     


Note: The radius from the centre of a circle to the point of tangency is perpendicular to the tangent line. So, from the figure, .Let be another tangent that is parallel to the first tangent So, from the figure below,

is the diameter of the circle.

The distance between parallel tangents to a circle is its diameter.


Equation of Tangent to a Circle in Parametric Form

The equation of a tangent to the circle at the point is


Equation of Tangent to a Circle in Slope Form

The equations of tangent to the circle of slope is as follows 

 & the point of contact is             


Note : 

  • There exist two tangents with same slope to a given circle.
  • Line touches the circle if
  • The equation of tangents of slope to the circle are obtained by the following replacements of and in the previous slope form equation: 

Hence, the slope form equation of tangents of slope to the circle becomes                                   

            Similarly, the coordinates of the point of contact becomes                                                

 Practice Problems on Tangent to a Circle at a Point 

Example : If a line is a tangent to the circle and it is perpendicular to a line where is tangent to the circle at the point , then

Answer:

 

Step 1 :

 

is tangent to the circle at the point

 

Hence, the equation of is

 


Step 2 :

 

Now, the line 

is perpendicular to

 

 

Equation of

Step 3 :

 

Now, is tangent to the second circle 

Perpendicular distance of from the centre of Radius of circle 

Centre

 

Hence, the correct option is

Example : The tangents to having inclination and intersect at If , then find the locus of

Answer:

Step 1 :

 

Given, circle

 

So, radius of the circle,

 

Centre of the circle

 

Let the coordinates of

 

Equation of the tangent of slope will be

 


Step 2 :

 

As point lies on the tangent,

 

The above equation is a quadratic equation in

Let the roots be and .

and

 

Step 3 :

 

But slope of the first tangent 

and slope of the second tangent

 

As

 

Now, replace with and with .

So, the locus of where (or) .

 

(because point of intersection of tangents must be out side the circle)

Example : If , then show that line touches a fixed circle, and then find the centre and radius of the circle.

Answer:

The condition for a line to be tangent to a circle is , where is the perpendicular distance from the centre of the circle to the tangent line and is the radius of the circle. 

Given,

Now, adding on both the sides, we get,

Taking square root on both sides, we get

This is in the form of perpendicular distance from the centre of the circle to the tangent line

Thus, centre and radius units

Example : If is a variable point on the circle with the centre at , and are the perpendicular from on the and axes, respectively, then show that the locus of the centroid of triangle is a circle with the centre at the centroid of triangle and radius equal to one-third of the radius of the given circle.

Answer:

 

Step 1 :

Given, the equation of the circle is

 

⇒ Centre ( and 

radius

Let

 

Let the locus of the centroid of be  

and

 

Now, after squaring and adding equations and , we get,

 

 

Finally the locus of the centroid of triangle is obtained by replacing with .

⇒ Centre and 

Radius


 

Step 2 :

 

Now, let the centroid of triangle be .

 

Therefore, we see that the locus of the centroid of triangle is a circle with the centre at the centroid of triangle and radius equal to one-third of the radius of the given circle.

Example : Find the general equation of the circle for which is a tangent.

Answer:

Given, tangent 

The equation of tangent to the circle at 

is given by

Comparing both the equations of tangents, we get 

Centre & radius,

So equation of the circle is

Now, the general equation of the circle is as follows :

FAQ’s on Tangent to a Circle at a Point 

Question 1.How many tangents can be drawn from a point inside the circle?

Answer: No tangent can be drawn from a point inside the circle.

Question 2.What is the relationship between the slopes of the tangent and normal to a circle at a point?

Answer: The product of the slopes of tangent and normal at a point on the circle is equal to

Question 3. If a chord and a tangent of a circle are at maximum distance, then what will be the limiting case of the chord?

Answer: The limiting case of the chord will be a tangent parallel to given one.

Question 4.Can two circles be tangent to each other?

Answer: Two circles are tangent to each other only if they touch each other at exactly one point.

Related Tangent to a Circle in Maths

NCERT Class 11 Maths Chapters

Sets Relations and Functions Triginometric Functions
Mathematical Induction Numbers and Quadriatic Equations Linear Inequalities
Premutations and Combinations Binomial Theorem Sequence and Series
Straight Lines Conic Sections 3 D Geometry
Limits and Derivatives Mathematical Reasoning Statistics
Probability  
 
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