Chapter 10 of class 10, Circles, talks about the various aspects of the circle. This topic is essential for exams; therefore, students must understand this chapter properly. This article focuses on the different aspects of circles, including the tangent of a circle, the radical axis, the normal line equation to the curve etc.

A tangent is a perpendicular line that touches the coordinates of the parameter. The tangent is perpendicular to the radius, which joins the centre to the boundary at the midpoint, also known as P. The point where the tangent meets the circle is known as the point of tangency. The mathematical equation for the tangent is Y= MX+c. X since the tangent is a straight line and Y is the point where the tangent meets the circle.

The specified equation for measuring a circle is r²= (x-a)² + (y-b)², where a and b are the circle's centres, and the r is the circle's radius. The equation points out the location of the centre of the circle and its radius. This way, the equation quickly determines the properties and measurements of the circle. In the equation, a, b and r are constants and have fixed values attached, while x and y are variables and do not have any particular value. 

 

Parametric Equation of a Circle is the equation where the coordinates of a circle and the circle's perimeter relate to each other.

The radical axis of the two circles is a line that connects two circles perpendicularly. In addition, there is also a radical axis to two disjoint circles, which are the locus of the points that create tangents with identical lengths in the two circles. Some properties of the radical axis are listed below:

1. The circles that do not contain a radical axis are concurrent.

2. The radical axis bisects the tangents of the circle. 

3. The straight line that connects the two circles is perpendicular to the radical axis. 

4. The radical axis and the standard chord are the same. 

5. A collective system of circles where every two circles have an identical radical axis is known as the coaxial system. 

6. The coaxial system of circles has one point on either side of the radical axis. These points are known as the limiting points of the radical axis of the circle. 

7. All circles passing through the limited points are orthogonal to the coaxial system. 

8. When two circles cut the third circle orthogonally, the radical axis of the two circles passes through the central point of the third circle.

The equation of the radical axis of the two circles is 2x(g1 -g2) + 2y (f1 – f2) + c1 – c2 = 0, where the straight lines connecting the two circles are at S1-S2=0.

The normal line equation is the standard line that passes through the parameter point. It is mainly perpendicular to the tangent of the circle. Therefore, to draw a normal line to the curve, it is essential to draw its tangent and join it to the intended point. 

 

The equation of the normal line to the curve and tangent of the circle is x²+y²+2gx+2fy+c=0. This formula brings out the values of the normal lines and the tangents of the circle since the tangent touches the curve at particularly one point, unlike normal lines that are perpendicular to the tangent lines. If the tangent lines form an angle with the x-axis in the positive direction, then the tangent's slope is tan θ. However, if the tangent slope is zero, then the value of the angle tan θ is also zero.

Here are some of the exercises-

Exercise 10.1

Q3.  A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at

a point Q so that OQ = 12 cm. Length PQ is :

(A) 12 cm

(B) 13 cm

(C) 8.5 cm

(D) √119 cm

Solution:

As per the above figure, you can see that the line coming from the centre of the circle to the tangent PQ is perpendicular to PQ.

Therefore, OP ⊥ PQ

By applying the pythagoras theorem in this triangle ΔOPQ you will get,

OQ² = OP²+PQ²

(12)² = 5²+PQ²

PQ² = 144-25

PQ² = 119

PQ = √119 cm

Therefore, the answer to this question is option D, i.e. √119 cm is the length of PQ.

Exercise 10.2

Q1. From point Q, the length of the tangent to a circle is 24 cm, and the distance of Q from the centre is 25 cm. What is the radius of the circle? 

(A) 7 cm

(B) 12 cm

(C) 15 cm

(D) 24.5 cm

Solution:

The first thing you need to do is draw a perpendicular from the centre O of the triangle to the point P on the circle touching the tangent. 

This line will be perpendicular to the tangent of the circle.

Therefore, OP is perpendicular to PQ, i.e. OP ⊥ PQ

From the above figure, it can be seen that △OPQ is a right angled triangle.

You already know as per the information that-

OQ = 25 cm and PQ = 24 cm

By using Pythagoras theorem in △OPQ,

OQ² = OP² +PQ²

(25)² = OP²+(24)²

OP² = 625-576

OP² = 49

OP = 7 cm

Therefore, the answer to this question is option A, i.e. 7 cm is the radius of the given circle.

1. What is a tangent? How many tangents does a circle contain?

A tangent is a perpendicular line that touches the coordinates of a parameter. However, it does not exceed the point it touches. There can be infinite tangents because a circle consists of an endless number of points. Every point of the circle has an equal distance with the circle's radius. Since the circumference has many points, there can be many targets in the circle. 

 

     2. How to draw a normal line equation to the curve?

It is essential to understand what a normal line equation is before drawing the normal line equation to the curve. A normal line equation is a line that touches the parameter point. It is also always perpendicular to the tangent line of the circle. Therefore, to draw the normal line equation to the curve, it is mandatory to draw the tangent line of the circle first. After drawing the tangent line, it becomes easier to draw the normal line to the curve since it is just a line perpendicular to the tangent drawn before. 

 

      3. Explain the equation of a circle?

The standard equation of a circle is r²= (x-a)² + (y-b)². In this equation, the letters a and b signify the circle's centres, and r signifies the circle's radius. However, this equation does not find out the area of the circle. Instead, it offers a way to find out the features and measurements of the circle. The values a, b and r are constants with specific values attached, while x and y are variables with no specific value attached and are just there for the equation. 

 

      4. What is the parametric equation of a circle?

The parametric equation of a circle is where the coordinates and the circle's parameters relate to each other. The parametric equation of the circle is x=acosθ, y=asinθ where the radius is r, and the circle's centre is O or 0. The angle formed by joining the centre and the parameter in the x-axis is angle θ which becomes the equation mentioned above.

 

The topic Circle is fundamental from the exam point of view. Numerous questions come from the topic. Therefore, the students need to learn this topic in detail so that they do not face any difficulty while attempting questions based on this topic in the exams. For this reason, students need to find a proper guide and study materials to grasp the topic quickly.