intersection of lines and circles
TRANSCRIPT
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Lines and CirclesBy: Winnie W. Poli
MT-I, MNHS
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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
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Lines and Circles
Lines & CirclesWinnie W. Poli
No Point of
Intersection
One Point of Intersection
Two Points ofIntersection
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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.
Lines & CirclesWinnie W. Poli
CAFEyDxCyAx
bmxy
,022
This system of 2 equations can be solved by
substitution.
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Problem 1
Where does the line y = -x + 1 intersect the circle
x2+ y2= 25?
Algebraic Solution: Graphical Solution (By Geogebra)
Lines & CirclesWinnie W. Poli
+ = 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).
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Problem 2
Show that the line
y = - x + 8 does notintersect the circle
x2+ y2= 25.
Lines & CirclesWinnie W. Poli
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Algebraic Solution: Problem 2
Show that the
line y = - x + 8
does notintersect the
circle x2+ y2=
25.
Lines & CirclesWinnie W. Poli
+ ( + 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.
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Answers
Lines & CirclesWinnie W. Poli
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)
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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.
Lines & CirclesWinnie W. Poli
d < 0, there is no point
of intersection
d = 0, there is one
point of intersection
d > 0, there are two
points ofintersection
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Lines & CirclesWinnie W. Poli
Problem 3
Determine the y- intercept b of the
line y = -x + b that is tangent to thecircle x2+ y2= 25.
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Algebraic Solution: Problem 3
Determine the
y- intercept
b of the liney = -x + b
that is
tangent to
the circle x2
+ y2= 25.
Lines & CirclesWinnie W. Poli
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
=
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PRACTICE
Lines & CirclesWinnie W. Poli
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
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Solution
If the line
y = -x + b
is tangent tothe circle
x2+ y2= 1,
what is thevalue of b?
Lines & CirclesWinnie W. Poli
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
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Solution
If the line
y = m (x4 )
is tangent tothe circle
x2+ y2= 1,
what is thevalue of m?
Lines & CirclesWinnie W. Poli
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
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Solution
For what
numbers m will
the line y = mxintersect the
circle
+
= 16?
Lines & CirclesWinnie W. Poli
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.
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Solution
Find the pointsof intersectionof the line that
contains
( 1, 4 ) & ( 3, 6 )with the circle
of radius2 13 and
center ( -3, -2)
Lines & CirclesWinnie W. Poli
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
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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.
Lines & CirclesWinnie W. Poli