How far have you gotten?
can help me solve this question ? thanks much !
show that ,for all values of p ,the point p given by x=ap^2 ,y=2ap lies on the curve y^2=4ax .
a)find the equation of the ormal to this curve at the point p.
If this normal meets the curve at the point Q (q^2,2aq) , show that p^2 +pq+2=0 .
b)determine the coordinates of R ,the point of intersection of the tangents of the curve at the point p and Q .
hence ,show that the line locus of the pint R is y^2(x+2a)+4a^3=0 .
Okay, I have worked the problem, but you MUST show some effort here before we can proceed. Also, I found a few typos in the problem statement. Here is how I feel the problem could be stated:
Show that, for all values of , the point lies on the curve .
a) Find the equation of the line normal to the curve at .
If this normal line also crosses the curve at , show that .
b) Determine the coordinates of ,the point of intersection of the tangents of the curve at the points and .
Hence ,show that the line locus of the point is .
I will be more than happy to help, but if you have no idea how (or any inclination to try) to eliminate the parameter to obtain the Cartesian equation, then I am really at a loss to help.
I will give you a nudge to begin. We have the parametric equations:
Here are two simple examples which may help you see how to eliminate a parameter.
1.) Suppose you are given the parametric equations:
Notice the equation for is linear, so we may easily solve for the parameter :
Now, if we substitute for into the equation for we get an equation that relates and where has been "eliminated." We then have a Cartesian equation:
2.) Suppose you are given the parametric equations:
Rather than solving one of the equations for , we may rewrite the equations as:
If we square both equations, we may then add and take advantage of the Pythagorean identity
Adding, we find:
Which we may write as: