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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:
|
Hence, the correct option is |
is tangent to the circle 
at the point 
is 
is perpendicular to
Equation of 
is tangent to the second circle 
Perpendicular distance of 
from the centre of 
Radius of circle 
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
|
|
|
Step 2 :
As point
The above equation is a quadratic equation in Let the roots be
|
Step 3 :
But and
As
Now, replace So, the locus of
(because point of intersection of tangents must be out side the circle) |
will be
lies on the tangent,
and 
.
and 
slope of the first tangent 
slope of the second tangent 
with 
and 
with 
.
where 
(or) 
.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 2 :
Now, let the centroid of triangle
Therefore, we see that the locus of the centroid of triangle |
and 
be 
and 
and 
, we get,
is obtained by replacing 
with 
.
and 
be 
.
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.
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