Mathematics

Question

B is any point on the curve y²=2x and A is a fixed point with coordinates (2,6). Find the equation of the curve trace by the mid-point of AB.

 

Ans: y²-6y-x+10=0

 

Thank you! :)

Answer 1

y^2 = 2x is a rightward opening parabola with vertex at the origin, (0,0).

"Midpoints" would include (2,4) and (10,6) the first is midway between (2,2) on the parabola y^2=2x and (2,6).

(10,6) is midway between (2,6) and (18,6) another point on the parabola y^2=2x

the curve connecting those 2 points is another parabola x=y^2-6y+10 = y^2-6y +9 + 1

or x-1=(y-3)^2 which has vertex at (1,3). Yet, the distance from (1,3) to (2,6) = sqr10, but the distance from (1,3) to the line y^2=2x is less than sqr2 That equation can't describe AB for any B on the parabola.

However, the initial problem assumes AB is from A to a point on the top side of the parabola y^2=2x, which ignores the bottom half of the parabola. There are problems with this problem