Parametric (1 Viewer)

[lyounamu]

The normal at P(2ap, ap^2) on the parabola x^2 = 4ay cuts the y-axis at Q and is producted to a point R such that PQ = QR.

i) given that te equation of the normal at P is x+py = 2ap + ap^3, find th ecoordinates of Q.

ii) Show that R has coordinates (-2ap, ap^2 + 4a)

iii) Show that the locus of R is a parabola and state its vertex.

I got Q i) and ii) but I am not sure if my answer for iii) is right. I just put i) and ii) for reference. Show me what you get in iii) Thanks.

 

[lolokay]

is the vertex (0, 4a), or am I way off?

x=-2ap, p = -x/2a
y = ap² + 4a
= x²/4a + 4a
dy/dx = x/2a = 0 when x = 0 (vertex)
y = 0 + 4a
so coordinates are (0,4a)

 

[lyounamu]

is the vertex (0, 4a), or am I way off?

x=-2ap, p = -x/2a
y = ap² + 4a
= x²/4a + 4a
dy/dx = x/2a = 0 when x = 0 (vertex)
y = 0 + 4a
so coordinates are (0,4a)

That's same as my answer. I am glad. I just rearranged the equation instead of finding the derivative but that's a good way, I reckon.