Coordinate Systems Help? Hyperbolae

Hello? Anyone there ?

Can someone help me with this question? I managed to do part i where i got
y=xp^2 -cp^3 +c/p
For the second part I equated the y from the original equation to the y in the equation for normal and now I'm stuck in big old mess.

The attachment isn't working for me.

im assuming your "big old mess" is correct, did you get as far as solving a quadratic using the quadratic formula?

Here's what I got so far:

looks okay yea, write it in the form $\displaystyle ax^2 + bx + c = 0$ and solve for x (youll need the quadratic formula and it might be tricky)

you should be expecting a quadratic here since the line crosses the curve(s) at 2 places, $\displaystyle (cp, \frac{c}{p})$ and the other coordinates they want you to find

I got (2cp , 3cp^3) and (-c/p^3 , -cp^3). Is that right? I'm not able to post my solution atm

you made a little slip on the LHS answer $\displaystyle x = \frac{2cp^4}{2p^3} = cp \neq 2cp$, otherwise thats correct

you should always notice in these type of questions (when it says to find the other coordinate) that one of the x values you get from the quadratic should be the original x value you were given, in this case $\displaystyle x = cp$ is a value from the quadratic which you used originally the find the equation of the line so it can be ignored since youre only concerned with the other coordinate

You solve the quadratic equation which gives the two points of intersection of the curve and the line. But one of them is going to be the point P since that's the point on the curve that the line is normal to.

You solve the quadratic equation which gives the two points of intersection of the curve and the line. But one of them is going to be the point P since that's the point on the curve that the line is normal to.

Which quadratic equation are you referring to? Is it new equation you get from equating the Cartesian to the tangent equation?

yea, the quadratic you solved to get the two x values, in this question one of the x values will be x=cp since you used the point P(cp, c/p) at the start to find the equation of the normal passing through P.

Its something you should be aware of as its a nice way to know whether the x values you find from the quadratic are correct.

So were the two lines intersect are the roots of the new equation