EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Baiq Zilalin Azzima Erma Sariwangi Febri Arianti

Fitria Windiarni

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

The circle is defined as the set of points within the same to a certain point. The same distance is called the radius of the circle and the point is called the center of the circle.

Conclusion:OA = OB = OC = OD = ...

CIRCLE DEFINITION

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

THE EQUATION OF CIRCLE WITH CENTRE and RADIUS The distance between two points and is T and radius

⟺ ⟺

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

EQUATION OF CIRCLE WITH CENTRE and RADIUS

So the equation of the circle with center M (a, b) and radius r is

(𝑥−𝑎)2+( 𝑦−𝑏)2=𝑟2

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

GENERAL EQUATIONS CIRCLE

Equation of a circle with a center point and radius r is

I, then the equation can be written as The general form of the equation of the circle is . This equation can be written as follows.

Note that this is the equation of a circle with center and radius

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Tangent line of circle is a line that cuts right circle at one point and perpendicular to the diameter of the circle passing through the point. PictureThe Common Tangent of Two Circles1. Inner Common Tangent Between The Two CirclesIn the picture, and is the inner common tangent of circle centered at and Q

TANGENT LINE DEFINITION

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

A

O

L

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

parallel to , then: Look at rectangle

, then: Then and Obtained:

radius circle centered at is ,Radius circle centered at is ,the length of inner common tangent is , the length of the line center (central) is .

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Triangle is right angled in point , so that or ,for So, the length of inner common tangent between the two circles is , : the length of inner common tangent : distance of the center of the first circle and the second circle : the radius of the first circle and the second circle

2. The Outer Common Tangent Between The Two CirclesIn the picture, and the outer common tangent of circle centered at and .

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Triangle is right angled in point , so that or , for  So, the length of the outer tangent between the two circles is ,  

parallel to Look at rectangle and , then:  Then and . 

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

: the length of outer common tangent : of the center of the first circle and the second circle : the radius of the first circle and the second circle

The Equation of The Tangent Line on a Circle

1. Equation of the Tangent Line on a Circle Through Point of P :a) Equation of the Tangent Line to the Circle with Center of

O(0,0) Through Point of P)

𝑥1 𝑥+𝑦1 𝑦=𝑟 2

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Proof :• h is tangent line through the

point of P• equation of a circle centered at

is • The line of h is perpendicular to

line of P, then • and

Since point of P(X1, y1) is on the circle of x2+y2 = r2, then satisfies :x1

2 + y12 = r2 ..... (2)

From (1) and (2) x1x + y1y = r2

h

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

b) The Equation of the Tangent Line to the Circle with Center of A(a,b) Pass through Point P

• MAP = • MAP .Mg = -1• Mg = - , substitution to

We get x1x – ax + ax1 + y1y – by + b y1 = x1

2 + y1

2 ......(*)

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Because P(x1, y1) have position at cicle L = (x – a)2 + (y – b)2 = r2,so that :(x1 – a)2 + (y1 – b)2 = r2

x12 + y1

2 = 2ax1 – a2 + 2by1 – b2 + r2 ..... (**)Subsitution (*) and (**) And we get the equation of tangent line trough P(x1, y1) is

2. The Equation of the Tangent Line to the Circle with Specific Gradient

a. The equation of the tangent line with gradient of to the circle with center of and radius of

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

b. The equation of the tangent line with gradient m to the circle with center P(a,b) and radius of r

Proff

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

3. The Equation Of The Tangent Line To The Circle And Passing Through An External Point

PQ and PR P on PQ , it means ; .... (1)P on PR, it means ;.... (2)Look at line g, it has equation x .... (3)

Line polar equation

Q(X2,X2)

R(X3,X3)

P(X1,X1)

g

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

POWER OF CIRCLE

Power And Length Of Tangent Line

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

POWER LINE

If given two circle L1 and L2 hence power line can look for. Taking example we will determine equation of radian power line of circle L1 = 2 1 11 and L2 22 2 22 and take example is point having same power to L1 and L2.According to 1

2 12 1 1 hence point power of P to circle of L1 is

P2 P

2 1P + 1P 1 and point power of P to circle of L1 is P2 P

2 2P + 2P 2

Point power of P to both circle is same so that :P

2 P2 1xP 1P 1P

2 P2 + 2P 2P 2

(a1 – a2)xP + (b1 – b2)yP + (c1 – c2) = 0If point of P run by hence obtained place domicile points having same power to circle of L1 and of L2 that is.

(a1 – a2)x + (b1 – b2)y + (c1 – c2) = 0Symbolically equation of radian power line of L1 = 0 and L2 = 0 written down as.L1 – L2 = 0

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Power Line is position of point which has the same power on two circles. Thus, there are some possibility:

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Power point

Suppose L1, L2, L3 are three circles that hubs are not on a straight line (concentric). The third circle has three power lines that intersect at one point. The third point of intersection of these lines is called the power point.

If three circles are concentric then its power lines parallel, and this means that the point of the third power of the circle is in the infinity point.

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Polar Line

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

From point T(x, y) maked tangents line at circle L: . If tangent points at the circle is S1(, ) dan S2 (', '). The tangent line equation at circle L with tangent points S1 and S2 is and . The tangent lines g1 and g2 through T so that the following equation applies

……………………. (i).………………….. (ii)

This is the equation of the line through the points of tangency are called S1 and S2 and straps arc tangent. Note that the equation of the tangent g bowstring same shape the equation of the circle tangent to L at the point of tangency T. Therefore that, regardless of the location of point T (inside, outside, or in a circle), then the equation point polar equation T the circle L: is

From the above description, obtained, if T outside the circle, then the line poles g a tangent bowstring.

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

Lets try !1. Determine the equation of a circle passing through the point

(0,4), (1, -1) and (1.3) !2. The following image is a cross section of 3 pieces of cake tins

place tubular radius 10 cm. Calculate the minimum length of rope to tie 3 pieces place the cake tin?

3. Given point of and the circle of .a). show that point is on the circleb) determine the equation of the tangent line to the circle passing through point of A

4. Find power and the length of point A(–1,4) on circle which its center on (2,–1) and radius 5 !

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE

EQUATION OF CIRCLE

THE TANGENT OF CIRCLE ON A POINT OR THROUGH A POINT ON THEIR EXTERIOR

POWER OF CIRCLE