In the graph at right there are five points of intersection

In the graph at right, there are five points of intersection among the line, the ellipse, and the hyperbola. Given the equations below, find the coordinates of points A through E. line: 2x - y = 0 ellipse: x^2 + 3y^2 - 12y = 0 hyperbola: x^2 - ey^2 + 6y = 0

Solution

2x-y = 0.....1

x² +3y² -12y = 0....2

From eq1

y = 2x

Putting value of y in eq 2

x² +3(2x)² -12*2x = 0

x² +12x² -24x = 0

13x² -24x = 0

x(13x-24) = 0

x = 0

13x-24 = 0

13x = 24

x = 24/13

For x = 0

2x - y = 0

y =0

For x = 24/13

2*24/13 = y

48/13 = y

B. (0,0)

D.(24/13, 48/13)

x² +3y² -12y = 0.....2

x² -3y² + 6y = 0....3

eq 2 - eq 3

6y² - 18y = 0

6y(y - 3) = 0

y = 3

x² +3y² -12y = 0.

x = sqrt(36 -27) = + 3 or -3

E.(3,3)

A.(-3,3)

x² -3y² + 6y = 0

2x - y = 0

y = 2x

x² -3(2x)² + 6(2x) = 0

x² -12x² + 12x = 0

- 11x² + 12x = 0

x(-11x +12) = 0

-11x = -12

x =12/11

y = 2*12/11 = 24/11

C(12/11, 24/11)