Hi, I've been trying to learn the C4 paper ahead of schedule and have gotten down to one last question I can't quite get.
Q5) A curve C has parametric equations $\displaystyle x = at^2$ , $\displaystyle y = 2at$. Show that the equation of the normal to C at the point P, whose parameter is $\displaystyle p$, is:
$\displaystyle px + y - 2ap - ap^3 = 0$
The normal to C at P meets the x-axis at Q. The perpendicular from P to the x-axis meets the x-axis at R. Find the length of QR.
This is my working so far:
$\displaystyle dx/dt = 2at$
$\displaystyle dy/dt = 2a$
$\displaystyle dy/dx = dy/dt * dt/dx = 1/t$ (Chain Rule)
Normal Gradient = $\displaystyle -1/m$ = $\displaystyle -p$
(Use p as parameter for normal equation)
$\displaystyle y - y1 = m(x - x1)$
$\displaystyle y - 2ap = -p(x - ap^2)$
$\displaystyle y - 2ap = -px + ap^3$
$\displaystyle px + y - 2ap - ap^3 = 0$ (As required)
And thats about as far as I got, I've been trying to let y = 0 and let q be the parameter, so:
$\displaystyle qx - 2aq - aq^3 = 0$
But do you then make them equal to each other and figure it out from there? I don't really know, its a WJEC paper too so I can't get any marking schemes without paying, so I thought this way would be better to get an understanding.
Thanks