Find a point on the line y = x that lies on the parabola whose focus is (0,2) and directrix is the x- axis. Describe the relationship between the line y = x and the parabola. Solution Axis of symmetry: x = 0 or y-axis Vertex = (0,0) a = distance from vertex to focus or to directrix on the axis of symmetry = - 2 Equation: x2 = 4ay x2 = -8y ...(1) Line y = x intersect the parabola x2 = -8y. so put y = x in x2 = -8y. then x2 = -8x or x2 + 8x = 0 x (x + 8)= 0 then x = 0 or x = -8 then y = x , so y = 0 or y = -8. then exist points is (0, 0) or (-8, -8)..

1. Find a point on the line y = x that lies on the parabola whose focus is (0,2) and directrix is the x-
axis. Describe the relationship between the line y = x and the parabola.
Solution
Axis of symmetry: x = 0 or y-axis
Vertex = (0,0)
a = distance from vertex to focus or to directrix on the axis of symmetry = - 2
Equation:
x2 = 4ay
x2 = -8y ...(1)
Line y = x intersect the parabola x2 = -8y.
so put y = x in x2 = -8y.
then x2 = -8x
or x2 + 8x = 0
x (x + 8)= 0
then x = 0 or x = -8
then y = x , so y = 0 or y = -8.
then exist points is (0, 0) or (-8, -8).