1. Circle: (i) A circle is the collection of those points in a plane that are at a given constant distance from a given fixed point in the plane. The fixed point is called the centre and the given constant distance is called the radius of the circle.  (ii) There is one and only circle passing through three given non-collinear points. 2. Position of a Point with respect to Circle: A point P lies inside or on or outside the circle c(O,r) according to as OP<r or OP=r or OP>r. 3. Circular Disc: The collection of all points lying inside and on the circle c(O,r) is called a circular disc with centre O and radius r. 4. Concentric Circles: Circles having the same centre and different radii are said to be concentric circles. 5. Arc of a Circle: (i) A continuous piece of a circle is called an arc of the circle. (ii) The length of an arc is the length of the fine thread which just covers it completely. (iii) Let c(O,r) be any circle. Then any angle whose vertex is O is called the central angle. (iv) A minor arc of a circle is the collection of those points of the circle that lie on and inside the arc central angle. (v) A major arc of a circle is the set of points of the circle that lie on or outside a central angle. (vi) The angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle. (vii) A diameter of a circle divides it into two equal parts which are arcs. Each of these two arcs is called a semi-circle. (viii) Angles in the same segment of a circle are equal. (ix) The angle in a semi-circle is a right angle. 6. Chord of a Circle and Theorems based on it: (i) A line segment joining any two points on a circle is called a chord of the circle. (ii) A chord passing through the centre of a circle is known as its diameter. (iii) If two arcs of a circle are congruent, then corresponding chords are equal. (iv) If two chords of a circle are equal, then their corresponding arcs are congruent. (v) The perpendicular from the centre of a circle to a chord bisects the chord. (vi) The line segment joining the centre of a circle to the mid-point of a chord is perpendicular to the chord. (vii) If two circles intersect in two points, then the line through the centres is perpendicular to the common chord. (viii) Equal chords of a circle are equidistant from the centre. (ix) Chords of a circle which are equidistant from the centre are equal. (x) Equal chords of a circle subtend equal angle at the centre. (xi) If the angles subtended by two chords of a circle at the centre are equal, the chords are equal. (xii) Of any two chords of a circle, the larger chord is nearer to the centre. (xiii) Of any two chords of a circle, the chord nearer to the centre is larger. 7. Congruent Circles: Two circles are said to be congruent if either of them can be superposed on the other so as to cover it exactly. 8. Cyclic Quadrilateral: (i) If all vertices of a quadrilateral lie on a circle, it is called a cyclic quadrilateral. (ii) The opposite angles of a cyclic quadrilateral are supplementary. (iii) If the sum of any pair of opposite angles of a quadrilateral is 180°, then it is cyclic. (iv) If one side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite angle. (v) An isosceles trapezium is cyclic.
Learn about the most important parts of a circle: the center, radii, diameters, circumference, and so on. Given two points $(x_1,y_1)$ and $(x_2,y_2)$, recall that their horizontal distance from one another is $\Delta x=x_2-x_1$ and their vertical distance from one another is $\Delta y=y_2-y_1$. (Actually, the word "distance'' normally denotes "positive distance''. $\Delta x$ and $\Delta y$ are signed distances, but this is clear from context.) The actual (positive) distance from one point to the other is the length of the hypotenuse of a right triangle with legs $|\Delta x|$ and $|\Delta y|$, as shown in figure 1.2.1. The Pythagorean theorem then says that the distance between the two points is the square root of the sum of the squares of the horizontal and vertical sides: $$ \hbox{distance} =\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+ (y_2-y_1)^2}. $$ For example, the distance between points $A(2,1)$ and $B(3,3)$ is $\sqrt{(3-2)^2+(3-1)^2}=\sqrt{5}$.
Figure 1.2.1. Distance between two points, $\Delta x$ and $\Delta y$ positive. As a special case of the distance formula, suppose we want to know the distance of a point $(x,y)$ to the origin. According to the distance formula, this is $\sqrt{(x-0)^2+(y-0)^2}=\sqrt{x^2+y^2}$. A point $(x,y)$ is at a distance $r$ from the origin if and only if $\sqrt{x^2+y^2}=r$, or, if we square both sides: $x^2+y^2=r^2$. This is the equation of the circle of radius $r$ centered at the origin. The special case $r=1$ is called the unit circle; its equation is $x^2+y^2=1$. Similarly, if $C(h,k)$ is any fixed point, then a point $(x,y)$ is at a distance $r$ from the point $C$ if and only if $\sqrt{(x-h)^2+(y-k)^2}=r$, i.e., if and only if $$ (x-h)^2+(y-k)^2=r^2. $$ This is the equation of the circle of radius $r$ centered at the point $(h,k)$. For example, the circle of radius 5 centered at the point $(0,-6)$ has equation $(x-0)^2+(y- -6)^2=25$, or $x^2+(y+6)^2=25$. If we expand this we get $x^2+y^2+12y+36=25$ or $x^2+y^2+12y+11=0$, but the original form is usually more useful.
Example 1.2.1 Graph the circle $x^2-2x+y^2+4y-11=0$. With a little thought we convert this to $(x-1)^2+(y+2)^2-16=0$ or $(x-1)^2+(y+2)^2=16$. Now we see that this is the circle with radius 4 and center $(1,-2)$, which is easy to graph. $\square$ Exercises 1.2Ex 1.2.1 Find the equation of the circle of radius 3 centered at: (answer) Ex 1.2.2 For each pair of points $A(x_1,y_1)$ and $B(x_2,y_2)$ find (i) $\Delta x$ and $\Delta y$ in going from $A$ to $B$, (ii) the slope of the line joining $A$ and $B$, (iii) the equation of the line joining $A$ and $B$ in the form $y=mx+b$, (iv) the distance from $A$ to $B$, and (v) an equation of the circle with center at $A$ that goes through $B$.
(answer) Ex 1.2.3 Graph the circle $x^2+y^2+10y=0$. Ex 1.2.4 Graph the circle $x^2-10x+y^2=24$. Ex 1.2.5 Graph the circle $x^2-6x+y^2-8y=0$. Ex 1.2.6 Find the standard equation of the circle passing through $(-2,1)$ and tangent to the line $3x-2y =6$ at the point $(4,3)$. Sketch. (Hint: The line through the center of the circle and the point of tangency is perpendicular to the tangent line.) (answer)
What is a circle? In geometry, a circle is the locus of points located at the same distance from a given fixed point. The word locus sounds complicated, but it means set of all points on a plane. The fixed point is called center of the circle. Another way then to define a circle is to say that it is the set of all points that are located at the same distance from a fixed point. Looking at the figure below, all the blue dots are located at the distance from the red dot (fixed point or center of the circle) and they form the locus of points mentioned above. Lines on a circle are given different names depending on how the line cuts the circle. There are five such lines.
A line segment joining two points on the circle is a chord. The following are examples of two chords. When a chord passes through the center, we call the chord a diameter. A diameter usually divides a circle into two equal halves and each half is called a semi-circle.
How to find the number pi
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What do you call the distance from a given fixed point to the set of points of a circle which is half of its diameter?
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