Intersection of Lines and Circles

8/13/2019 Intersection of Lines and Circles

1/18

Lines and CirclesBy: Winnie W. Poli

MT-I, MNHS


2/18

Intersection of Lines

Consider two lines L1and L2

Do L1and L2 always have a point of

intersection?When will they have a point of

intersection?

How do you find the point of intersection

if there exists ?

Lines & CirclesWinnie W. Poli


3/18

Lines and Circles


No Point of

Intersection

One Point of Intersection

Two Points ofIntersection


4/18

Finding the Point of Intersection

Solve a system of two equations, one of which

is linear in x and y and the other is quadratic

in x and y.


CAFEyDxCyAx

bmxy

,022

This system of 2 equations can be solved by

substitution.


5/18

Problem 1

Where does the line y = -x + 1 intersect the circle

x2+ y2= 25?

Algebraic Solution: Graphical Solution (By Geogebra)


+ = 25 + ( + 1)= 25 by subs.

+ 2 + 1 25 = 0

2 2 24 = 0

12 = 0

4 + 3 = 0 = 4 , = 4 + 1 = 3

= 3 , = 3 + 1 = 4

The points of intersection are

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


6/18

Problem 2

Show that the line

y = - x + 8 does notintersect the circle

x2+ y2= 25.



7/18

Algebraic Solution: Problem 2

Show that the

line y = - x + 8

does notintersect the

circle x2+ y2=

25.


+ ( + 8)

= 25

+ 16 + 64 25 = 0

2

16x + 39 = 0

=(16) (16)4(2)(39)

2(2)

=16 256 312

2(2)

Hence, there is no point of intersection.


8/18

Answers


x2+ y2= 5 & y = -x + 3

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

x2

+ y2

= 11 & =

x2+ y2-4x -6y + 9 = 0 & 2x + y = 5

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

(2,1) & (1,2)

No pt. of intersection

(

,

) & (2,1)

(

,

) & (

,

)

(3,0)


9/18


10/18

Determining the Number of

IntersectionThe discriminant inthe quadratic

equation that willresult after thesubstitution will tellhow many points of

intersection there willbe between the lineand the circle.


d < 0, there is no point

of intersection

d = 0, there is one

point of intersection

d > 0, there are two

points ofintersection


11/18


Problem 3

Determine the y- intercept b of the

line y = -x + b that is tangent to thecircle x2+ y2= 25.


12/18

Algebraic Solution: Problem 3

Determine the

y- intercept

b of the liney = -x + b

that is

tangent to

the circle x2

+ y2= 25.


y = -x + b is tangent to the circle x2+y2= 25.

= 0

x2

+ (-x + b)2

= 25x2+ x2 -2bx + b2 = 25

2 x2 - 2bx + b225 = 0

(2)

4 2

25 = 04 8 + 200 = 0 + 50 = 0

=


13/18

PRACTICE


If the line y = -x + b is tangent to the circle

x2+ y2= 1, what is the value of b?

If the line y = m (x

4 ) is tangent to the circlex2+ y2= 1, what is the value of m?

For what numbers m will the line y = mx intersect

the circle + = 16?

Find the points of intersection of the line that

contains ( 1, 4 ) and ( 3, 6 ) with the circle of radius

2 13and center ( -3, -2)

1

2

3

4


14/18

Solution

If the line

y = -x + b

is tangent tothe circle

x2+ y2= 1,

what is thevalue of b?


The line y = -x + b is tangent to

the circle x2+ y2= 1

There is 1 pt. of intersection

d = 0

+ ( + )1 = 0

+ 2 + 1 = 0

= (2) 4(2)( 1) = 04 8 + 8 = 0

= 2

1


15/18

Solution

If the line

y = m (x4 )

is tangent tothe circle

x2+ y2= 1,

what is thevalue of m?


The line y = m (x

4 ) is tangent to thecircle x2+ y2= 1

There is 1 pt. of intersection

d = 0

+ ( 4)1 = 0

+ 8 + 16 1 = 0

= (8) 4(+1)(16 1)= 0

64 64 60 + 4 = 0

15 + 1 = 0

=

2


16/18

Solution

For what

numbers m will

the line y = mxintersect the

circle

+

= 16?


The line y = mx intersects the

circle x2+ y2= 16

The points of intersection is

either 1 or 2

0

+ 16 = 0

= 0 4( + 1)(16) 064( + 1) 0 ( )

Therefore , m is any real number.

3


17/18

Solution

Find the pointsof intersectionof the line that

contains

( 1, 4 ) & ( 3, 6 )with the circle

of radius2 13 and

center ( -3, -2)


line containing ( 1, 4 ) & ( 3, 6 ) 4

1=

6 4

3 1= 1

= + 3

Circle + + 6 + 4 + 9 + 4 = 52

Pts. of intersection

+ ( + 3)+6 + 4( + 3) 39 = 0

+ + 6 + 9 + 6 + 4 + 12 39 =0

2 + 16 18 = 0 + 8x 9 = 0

(, ) & (,)

4


18/18

Assignment

Points (-2, 2) & (-3, 1)

are on the circle

x2-2x + y2+4y = 20.

Tangents to the circle

are drawn at the given

points. Find the point

of intersection of thetwo tangent lines.


Documents

Intersection of Lines and Circles