Introduction to circle

Circle is defined a locus of a point whose
distance from a fixed point = constant

In this definition of circle,

• the constant distance is called radius.
• the fixed point is called center

Parts of Circle

Center

The fixed point from which the distance to circumference is constant is called center .
O is Center

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Radius

A segment drawn from the center to the circumference is a called radius.
OA is Radius

Circumference

The locus of point which is equidistant from center is called circumference.
The locus is circumference

Diameter

A segment drawn through the center and ends to the circumference is called diameter.
AB is diameter

Chord

A segment that joins any two points on the circumference is called segment. Diameter is the longest chord.
AB is a Chord

Arc

A continuous piece of circumference is called an arc. There are two types of arc: Major (greater than half) and Minor (less than half). An arc half to the circle is called semi-circle.
AB (clockwise) is majorr Arc, AB (anti-clockwise) is minor Arc

Sector

A region enclosed by an arc with two radii and joining to the center is called sector. There are two types of sector: Major (greater than half) and Minor (less than half). A sector half to the circle is called semi-circle.
AB(clockwise) is major Sector and AB(anti-clockwise) is minor Sector

Segment

A region enclosed by an arc with a chord is called segment. There are two types of segment: Major (greater than half) and Minor (less than half). A segment half to the circle is called semi-circle.
AB (anti-clockwise) is minor segment

Tangent

A straight line that touches the circle exactly once but does not cut it, however extended, is called tangent.

Secant

A straight line that cuts a circle at two points. It is extension of chord.

Central Angle

An angle made at the center of a circle is called central angle.

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Equation of circle

1. center at origin: \(x^2+y^2=r^2\)
   center= (0,0)
   radius= r
   
2. center at any point (h,k): \((x-h)^2+(y-k)^2=r^2\)
   center= (h,k)
   radius= r
   
3. general equation of circle : \(x^2+y^2+2gx+2fy+c=0\)
   center= (-g,-f)
   radius= \(\sqrt{g^2+f^2-c} \)
4. If two end points \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) of a diameter of a circle is given, then
   Method 1
   
   1. Find the distance between the endpoints \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) given by
      \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
   2. Find the radius of circle given by
      \(r= \frac{1}{2} \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
   3. Find the center of the circle given by
      \(h= \frac{x_1+x_2}{2}\)
      \(k= \frac{y_1+y_2}{2}\)
   4. Find the equation of the circle
      \((x-h)^2+(y-k)^2=r^2\)
      \((x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0\)

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   Method 2
   If two end points \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) of a diameter of a circle is given, then
   1. take any arbitrary point on the circle given by
      \(P=(x,y)\)
   2. Find slope of both lines at P
      \(m_1= \frac{y-y_1}{x-x_1} \)
      \(m_2= \frac{y-y_2}{x-x_2} \)
   3. Find the center of the circle given by
      \(m_1.m_2=-1\)
      \((x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0\)

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Partcular case of Circle

Let center equation of a circle be \((x-h)^2+(y-k)^2=r^2\) in which
center= (h,k)
radius= r
Then

1. Equation of the circle touching the x-axis
   \((x-h)^2+(y-k)^2=k^2\)
2. Equation of the circle touching the y-axis
   \((x-h)^2+(y-k)^2=h^2\)
3. Equation of the circle touching the both axis
   \((x-h)^2+(y-h)^2=h^2\)