Tangents Normals and parameters

[Peter Pedant]

A curve is given by x=t2, y=t3. Find the eqn of the tangent with parameter u and the eqn of the normal to the curve at a pt whose parameter is v

Find the parameter of the point where the tangent at the pt with the parameter( 2/3)rt2= two thirds[ root two] meets the curve again and show that it is normal there. A level 1968

[Peter Pedant]

Answer implicitly differentiate and get dy/dx is 3t/2.

So I get the tangent in the usual way with parameter u is y=3ux/2 – u3/2

And the normal with parameter v is y = -2x/3 + (v3 – 2v/3)

And so the tangent with parameter two-thirds root 2 is
y = rt2.x -16(rt2)/27

And what you do is solve this simultaneously with x=t2, y=t3
Jiggle it around you should get magically an eqn which looks like the Normal with say parameter w


In fact putting in t3 and t2 into the tangent eqn – I get an insoluble cubic
Any ideas ? thanks

[at least I try to solve the q before putting it on A/bank]

[factor-fiction]

Can I check the question. I have tried to insert the square and cube symbols and a square root symbol.
Please confirm I have understood the question.
I'll then have a look later tonight

A curve is given by x=t², y=t³. Find the equation of the tangent with parameter u and the equation of the normal to the curve at a point whose parameter is v

Find the parameter of the point where the tangent at the point with the parameter ( 2/3)√2 meets the curve again and show that it is normal there. A level 1968

[Peter Pedant]

thank you
yes that is the one exactly
many thanks for having a look

the constant I get for the tangent is u-cubed over two

the constant I get for the normal is v-cubed minus two vee over three

[vascop]

I have solved this and I will post again later with a link to my googledocs solution.

[vascop]

@Peter: Here is the link to my solution:
https://docs.google.c...0Y2QtMTcwZTJjMTljNTRm
Let me know what you think.
Vasco

[Peter Pedant]

Incredible !

I missed the significance of three real roots and two have to be repeated completely. Many thanks

PP

[vascop]

@Peter: Just made a few changes to correct typos and a little alternative approach in square brackets:
https://docs.google.c...2ZmItZWFlNzQzYjQxMDcz