Precalculus Exploration 4: Lines and Circles

(Export results to html for the students to view later.) Visual exploration of the combined plot of a line and a circle, in which each student first volunteers to explain to the class why each graph is correct, and then how to find any points of intersection or show algebraically that they don't exist. As we scroll through the document with the class, only the equations and the plot are displayed initially.

`Exercise 19. Graph the following line and circle, and then find any points of intersection.`

y = 2*x+8, `  `*(x+3)^2+(y-2)^2 = 5

[Maple Plot]

`Substitute: `*y = 2*x+8, ` into `*((x+3)^2+(y-2)^2) = 5

(x+3)^2+(2*x+6)^2 = 5

5*x^2+30*x+45 = 5

5*x^2+30*x+40 = 0

x = -2, -4, ` `*y = 4, 0

`Intersection Points: `*[x[1], y[1]] = [-2., 4.], [x[2], y[2]] = [-4., 0.]

`Exercise 27. Graph the following line and circle, and then find any points of intersection.`

y = -x+23, `  `*x^2+y^2 = 289

[Maple Plot]

`Substitute: `*y = -x+23, ` into `*(x^2+y^2) = 289

x^2+(-x+23)^2 = 289

2*x^2-46*x+529 = 289

2*x^2-46*x+240 = 0

x = 8, 15, ` `*y = 15, 8

`Intersection Points: `*[x[1], y[1]] = [8., 15.], [x[2], y[2]] = [15., 8.]

`Exercise 28. Graph the following line and circle, and then find any points of intersection.`

y = 8/9*x+10/9, `  `*x^2+y^2 = 100

[Maple Plot]

`Substitute: `*y = 8/9*x+10/9, ` into `*(x^2+y^2) = 100

x^2+(8/9*x+10/9)^2 = 100

145/81*x^2+160/81*x+100/81 = 100

145/81*x^2+160/81*x-8000/81 = 0

29*x^2+32*x-1600 = 0

x = 200/29, -8, ` `*y = 210/29, -6

`Intersection Points: `*[x[1], y[1]] = [6.90, 7.24], [x[2], y[2]] = [-8., -6.]

`Exercise 29. Graph the following line and circle, and then find any points of intersection.`

y = 2*x-7, `  `*x^2+y^2 = 7

[Maple Plot]

`Substitute: `*y = 2*x-7, ` into `*(x^2+y^2) = 7

x^2+(2*x-7)^2 = 7

5*x^2-28*x+49 = 7

5*x^2-28*x+42 = 0

x = 14/5-1/5*I*14^(1/2), 14/5+1/5*I*14^(1/2)

`No Intersection Points.`

`Exercise 30. Graph the following line and circle, and then find any points of intersection.`

y = -1/2*x+5, `  `*x^2+y^2 = 20

[Maple Plot]

`Substitute: `*y = -1/2*x+5, ` into `*(x^2+y^2) = 20

x^2+(-1/2*x+5)^2 = 20

5/4*x^2-5*x+25 = 20

5/4*x^2-5*x+5 = 0

x^2-4*x+4 = 0

x = 2, 2, ` `*y = 4, 4

`Intersection Point: `*[x[1], y[1]] = [2., 4.]

`Exercise 31. Graph the following line and circle, and then find any points of intersection.`

y = -5/2*x-1/2, `  `*x^2+y^2 = 169

[Maple Plot]

`Substitute: `*y = -5/2*x-1/2, ` into `*(x^2+y^2) = 169

x^2+(-5/2*x-1/2)^2 = 169

29/4*x^2+5/2*x+1/4 = 169

29/4*x^2+5/2*x-675/4 = 0

29*x^2+10*x-675 = 0

x = -5, 135/29, ` `*y = 12, -352/29

`Intersection Points: `*[x[1], y[1]] = [-5., 12.], [x[2], y[2]] = [4.66, -12.1]

`Exercise 32. Graph the following line and circle, and then find any points of intersection.`

y = 5, `  `*(x-2)^2+(y-3)^2 = 4

[Maple Plot]

`Substitute: `*y = 5, ` into `*((x-2)^2+(y-3)^2) = 4

(x-2)^2+4 = 4

x^2-4*x+8 = 4

x^2-4*x+4 = 0

x = 2, 2, ` `*y = 5, 5

`Intersection Point: `*[x[1], y[1]] = [2., 5.]

`Exercise 33. Graph the following line and circle, and then find any points of intersection.`

y = 3^(1/2)*x, `  `*x^2+(y-4)^2 = 16

[Maple Plot]

`Substitute: `*y = 3^(1/2)*x, ` into `*(x^2+(y-4)^2) = 16

x^2+(3^(1/2)*x-4)^2 = 16

4*x^2-8*3^(1/2)*x+16 = 16

4*x^2-8*3^(1/2)*x = 0

x^2-2*3^(1/2)*x = 0

x = 2*3^(1/2), 0, ` `*y = 6, 0

`Intersection Points: `*[x[1], y[1]] = [3.46, 6.], [x[2], y[2]] = [0., 0.]

`Exercise 34. Graph the following line and circle, and then find any points of intersection.`

y = x-3, `  `*(x-5)^2+(y+2)^2 = 16

[Maple Plot]

`Substitute: `*y = x-3, ` into `*((x-5)^2+(y+2)^2) = 16

(x-5)^2+(x-1)^2 = 16

2*x^2-12*x+26 = 16

2*x^2-12*x+10 = 0

x^2-6*x+5 = 0

x = 1, 5, ` `*y = -2, 2

`Intersection Points: `*[x[1], y[1]] = [1., -2.], [x[2], y[2]] = [5., 2.]