Coordinate Geometry: Finding the Locus of a Midpoint on a Curve

[Michael_Light]

Homework Statement

A variable point P lies on the curve y² = x³ and is joined to a fixed point A with coordinate (2,0). Prove that the locus of the mid-point of AP is y²= 2(x-1)³.

Homework Equations

The Attempt at a Solution

According to what i know, I need to know the parameter for the curve y² = x³ to prove it, but what is the parameter for the curve y² = x³? Can anyone guide me?

 

[Curious3141]

Homework Statement

A variable point P lies on the curve y² = x³ and is joined to a fixed point A with coordinate (2,0). Prove that the locus of the mid-point of AP is y²= 2(x-1)³.

Homework Equations

The Attempt at a Solution

According to what i know, I need to know the parameter for the curve y² = x³ to prove it, but what is the parameter for the curve y² = x³? Can anyone guide me?

Start with a generalised point on the curve. Let the x-coordinate of the point be t. What will the y-coordinate be in terms of t?

Now work out the midpoint of the segment AP. This is as simple as taking the average of the coordinates of the point of the curve and the fixed point (2,0).

You now have the co-ordinates of the general midpoint. This is a parametric equation in terms of t. Undo the parametrisation to express the y-coord. directly in terms of x, simplify and you should quickly have the result.