Lines and Sectors

By: Winnie W. Poli

MT-I, MNHS

Intersection of Lines

Consider two lines L1 and L2

Do L1 and L2 have always a point of

intersection?

When will they have a point of

intersection?

How do you discover the point of intersection

if perhaps there exists?

Lines & Groups

Winnie W. Poli

Lines and Circles

Two Points of

Intersection

Simply no Point of

Intersection

1 Point of Intersection

Lines & Groups

Winnie Watts. Poli

Choosing the Point of Intersection

• Solve a method of two equations, one among which

is usually linear in x and y as well as the other is quadratic

in x and y.

y m x m

Ax Cy Dx Ey F 0, A C

2

2

• This technique of 2 equations can be resolved by

alternative.

Lines & Circles

Winnie W. Agente

Problem 1

Where will the line sumado a = -x + one particular intersect the circle

x2 + y2 = twenty-five?

Algebraic Solution: Graphical Answer (By Geogebra)

𝑥 2 + 𝑦 2 = 25

𝑥 2 + (−𝑥 + 1)2 sama dengan 25 simply by subs.

𝑥 2 & 𝑥 two − 2𝑥 + 1 − twenty-five = 0

2𝑥 two − 2𝑥 − 24 = zero

𝑥 2 − 𝑥 − doze = 0

𝑥−4 𝑥+3 =0

𝑥 = four, 𝑦 = −4 & 1 sama dengan −3

𝑥 = −3,

𝑦 =3+1=4

The parts of intersection happen to be

(4, -3) and (-3, 4).

Lines & Sectors

Winnie Watts. Poli

Problem 2

Demonstrate that the collection

y = - times + almost eight does not

intersect the group of friends

x2 & y2 sama dengan 25.

Lines & Sectors

Winnie W. Poli

Algebraic Solution: Difficulty 2

Demonstrate that the

series y sama dengan - times + almost 8

does not

intersect the

group x2 + y2 =

25.

two

2

𝑥 + (−𝑥 + 8) = twenty-five

𝑥 a couple of + 𝑥 2 − 16𝑥 & 64 − 25 sama dengan 0

2𝑥 2 − 16x & 39 sama dengan 0

−(−16) ± (−16)2 −4(2)(39)

𝑥=

2(2)

sixteen ± 256 − 312

𝑥=

∉ℝ

2(2)

Therefore, there is no level of area.

Lines & Circles

Winnie W. Poli

Answers

x2 + y2 = your five & sumado a = -x + three or more

x2 & y 2 = four

x2

+

y2

& y = -x & 4

-4x -6y + 9 sama dengan 0 &

x2 & y2 sama dengan 11

&

(2, 1) & (1, 2)

Not any pt. of intersection

2 21

(, )

your five 5

two times + con = a few

𝑦=

one particular

2

x2 + y2 -6x & 9 = 0 & x+ con = 3

Lines & Circles

Winnie W. Agente

(

40 1

, )

2

a couple of

& (2, 1)

& (−

(3, 0)

40 1

, )

2

2

Practice Physical exercises

• Discover the point or perhaps points of intersection, if virtually any,

of the provided line and circle.

1 . x2 + y2 = 5 & y sama dengan -x + 3

2 . x2 + y2 = 4 & y = -x + 4

a few. x2 & y2 -4x -6y + 9 = 0 & 2x & y sama dengan 5

one particular

2 & y2 sama dengan 11

5. x

&

y

a couple of

5. x2 + y2 -6x + 9 sama dengan 0 & x+ con = 3

Lines & Circles

Winnie W. Poli

Determining the quantity of

Intersection

The discriminant in

the

quadratic

equation which will

result

following

the

substitution will inform

how various points of

area there will

end up being between the range

and the circle.

d < 0, there is absolutely no point

of intersection

d = zero, there is 1

point of intersection

deb > zero, there are two

points of

area

Lines & Circles

Winnie W. Agente

Problem three or more

Determine the y- intercept b with the

line sumado a = -x + m that is tangent to the

ring x2 + y2 sama dengan 25.

Lines & Groups

Winnie Watts. Poli

Algebraic Solution: Trouble 3

y sama dengan -x + b can be tangent towards the circle x2 +

y2 = 25.

Determine the

y- intercept 𝑑 = zero

b of the line x2 + (-x & b)2 = 25

con = -x + w

x2 & x2 -2bx + b2 = 25

that is

a couple of x2 -- 2bx + b2 – 25 sama dengan 0

tangent to

(−2𝑏)2 −4 two 𝑏2 − 25 sama dengan 0

the circle x2

+ y2 = 25.

4𝑏 a couple of − 8𝑏 2 & 200 sama dengan 0 → −𝑏 two + 50 = 0

𝒃 = ±𝟓 𝟐

Lines & Circles

Winnie W. Agente

PRACTICE

you

If the range y = -x & b is usually tangent for the circle

x2 + y2 = one particular, what is the importance of b?

a couple of

If the collection y = m (x – 4 ) is tangent for the circle

x2 + y2 = you, what is the cost of m?

several

For what amounts m will the line y = mx intersect

the circle 𝑥 2 & 𝑦 a couple of = of sixteen?

4

Get the points of intersection of the line that

contains ( 1, some ) and ( several, 6 ) with the circle of radius

2 13and center ( -3, -2)

Lines & Circles

Winnie W. Agente

Solution

1

If the series

y sama dengan -x & b

is definitely tangent to

the ring

x2 & y2 = 1,

precisely what is the

worth of w?

• The queue y = -x & b can be tangent to

the group of friends x2 + y2 sama dengan 1

• There is one particular pt. of intersection

• d=0

𝑥 2 & (−𝑥 & 𝑏)2 −1 = zero

𝑥 a couple of + 𝑥 2 − 2𝑏𝑥 & 𝑏 a couple of − 1 = zero

𝑑 sama dengan (−2𝑏)2 − 4(2)(𝑏 2 − 1) = 0

4𝑏 two − 8𝑏 2 & 8 sama dengan 0

𝑏=± 2 ...