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 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”.
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.
The equation of a tangent to the circle at the point is
The equations of tangent to the circle of slope is as follows
& the point of contact is
Hence, the slope form equation of tangents of slope to the circle becomes
Similarly, the coordinates of the point of contact becomes
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
Hence, the correct option is
Example : The tangents to having inclination and intersect at If , then find the locus of
Step 1 :
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 .
Step 3 :
But slope of the first tangent
and slope of the second tangent
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.
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.
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.
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.
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 :
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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