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Circles

Basics of Circle

You might already know that Circles have the least surface area but did you ever think about the geometry which makes it so? Well yes, like straight lines, circles have geometry too and trust me when I say, it is as easy as its linear counterpart. After all, how difficult is it to understand a curve. So, let’s roll around the concepts and understand a Circle.

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Introduction to Circles
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Equation of Circle –

Usually, just after the above heading, you’ll find the line written “The equation of a circle is given by…!”. But let’s not do that. We will derive it from scratch.

Let us take a circle of radius ‘r’ and put it on the coordinate plane. Take the centre at O(h,k) and then a point A on the circumference and assume its coordinates to be (x,y) and join OA. Now, we will assume a point B inside the given circle such that OB is ⊥ to AB which gives us a right angled triangle AOB with right angle at B. This makes OA, hypotenuse.

Circle Basics

From Pythagoras Theorem, we can say that

\( OB^2 + AB^2 = OA^2 \)

Here, OA = r; OB = difference in the x-coordinate = x – h; AB = difference in the y-coordinate = y – k

which means, \( (x-h)^2 + (y-k)^2 = r^2 \)

and this is the equation of a circle. Now if you’re thinking that this is just at one point, then stop yourself right there. You can take a similar point on any part of the circumference and still you’ll get the same result.

So, this equation is more like a collection of all such points on the circumference and is, hence, the equation of a circle.

The centre of this circle is at (h, k) and if you move it to the origin then the equation will become

 \( x^2 + y^2 = r^2 \)

Equation of circle in parametric form –

Parametric Equation of circle with centre \((h,k)\) and radius R is given by

\( x=h+R \cos{\theta}\) & \( y=k+R \sin{\theta} \)

where θ is the parameter.

Solved Examples for You

Q. 1 Convert the equation \( x^2 + y^2 – 4x + 6y – 12 = 0  \) into standard form and hence find its centre.

Sol – Here we have \( x^2 + y^2 – 4x + 6y – 12 = 0 \)

⇒ \( x^2 – 4x + y^2 + 6y – 12 = 0 \)

Add & subtract 4 & 9 to get perfect squares,

⇒  \( x^2 – 4x + 4 + y^2 + 6y + 9 – 25 = 0 \)

⇒  \( (x – 2)^2 + (y + 3)^2 = 25 \)

⇒  \( (x – 2)^2 + (y – (-3))^2 = 5^2 \)

which is the standard form of equation of circle. Also, the centre is at (2, -3).

Q.2: If we have a circle of radius 20 cm with its centre at the origin, the circle can be described by the pair of equations?

Sol – We know that for parametric form of equation of circle,

\( x=h+R \cos{\theta}\) & \( y=k+R \sin{\theta} \)

Here, since the centre is at (0, 0), so, h = k = 0 and it is already mentioned that radius is 20 cm. Thus, \(x=20 \cos{t}\) & \(y=20 \sin{t}\) are the required pair of equations.

Q. What is a circle in math?

A circle is referred to as the locus of all points intermediate from a central point. In other words, all the points are equally distant from the central point in a circle. Moreover, it consists of radius, diameter and circumference as well.

Q. What is 90 degrees in a circle?

A. First, let us make clear that a circle divides into 360 equal degrees. Thus, this means that a right angle is 90°. So, the angle of an equilateral triangle comprises 60 degrees. Then again, scientists, engineers, and mathematicians usually measure angles in radians.

Q. What are the properties of a circle?

A. When we look at the essential properties of a circle, we see that it has many. A circle is congruent when it had equal radii. Moreover, looking at the diameter of the circle, we see that it is the longest chord of a circle. Further, equal chords and equal circles have the equal circumference.

Q. What is 2π?

A full circle comprises of 2πradians (approximately 6.28). Moreover, always remember that an arc of a circle is what defines a radian. Further, the length of the arc is equal to the radius of a circle.

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