Intersection of Lines and circles

Lines and Circles
By: Winnie W. Poli
MT-I, MNHS

Intersection of Lines
Consider two lines L1 and L2
Do L1 and 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 & Circles
Winnie W. Poli

Lines and Circles
Two Points of
Intersection

No Point of
Intersection

One Point of Intersection
Lines & Circles
Winnie W. Poli

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.

y  m x b
Ax  Cy  Dx  Ey  F  0, A  C
2

2

• This system of 2 equations can be solved by substitution. Lines & Circles
Winnie W. Poli

Problem 1
Where does the line y = -x + 1 intersect the circle x2 + y2 = 25?
Algebraic Solution: Graphical Solution (By Geogebra)
? 2 + ? 2 = 25
? 2 + (−? + 1)2 = 25 by subs.
? 2 + ? 2 − 2? + 1 − 25 = 0
2? 2 − 2? − 24 = 0
? 2 − ? − 12 = 0
?−4 ?+3 =0
? = 4 , ? = −4 + 1 = −3
? = −3 ,
? =3+1=4
The points of intersection are
(4, -3) and (-3,4).

Lines & Circles
Winnie W. Poli

Problem 2
Show that the line y = - x + 8 does not intersect the circle x2 + y2 = 25.

Lines & Circles
Winnie W. Poli

Algebraic Solution: Problem 2
Show that the line y = - x + 8 does not intersect the circle x2 + y2 =
25.

2

2

? + (−? + 8) = 25
? 2 + ? 2 − 16? + 64 − 25 = 0
2? 2 − 16x + 39 = 0
−(−16) ± (−16)2 −4(2)(39)
?=
2(2)
16 ± 256 − 312
?=
∉ℝ
2(2)
Hence, there is no point of intersection.
Lines & Circles
Winnie W. Poli

Answers x2 + y2 = 5 & y = -x + 3 x2 + y 2 = 4

x2

+

y2

& y = -x + 4

-4x -6y + 9 = 0 &

x2 + y2 = 11

&

(2,1) & (1,2)
No pt. of intersection
2 21
( , )
5 5

2x + y = 5
?=

1
2

x2 + y2 -6x + 9 = 0 & x+ y = 3
Lines & Circles
Winnie W. Poli

(

42 1
, )
2
2

& (2,1)

& (−
(3,0)

42 1
, )
2
2

Practice Exercises
• Find the point