distance question?

Question

Consider the parabola y = x^2
a point P(x,y) lies on the parabola

d is the distance between P and (0,1)

express the distance d as a function of the y coordinate of P

Answer 1

d ² = x ² + (y - 1) ²
d ² = y + (y - 1) ²
d ² = y + y ² - 2 y + 1
d ² = y ² - y + 1
d = √ (y² - y + 1)

Answer 2

This is the same as finding the distance between any two points. You simply need to take the difference between the x and y coordinates. In this case, you have two points. The first point is (x, y) = (0, 1) and the second point is (x, y) = ( sqrt(y), y) because we know that y = x^2, which is the same as x = sqrt(y).

Answer 3

The distance of P(x,y) from (0,1) is sqrt((x-0)^2 + (y-1)^2)

If y=x^2

Then d = sqrt((x-0)^2 + (y-1)^2) =
= sqrt((x-0)^2 + (x^2 - 1)^2) =
= sqrt(x^2 + (y - 1)^2) = sqrt(y + y^2 - 2y + 1) =
= sqrt(y^2 - y + 1)

Answer 4

d= sqrt(x^2 +(y-1)^2)
But y = x^2
So d = sqrt(y +(y-1)^2)
d= sqrt(y^2-y+1)