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Got another Q here. P(2at,at^2) and Q(2as,as^2) are two points on the parabola x^2=4ay . The chord PQ subtends a right angle at the origin. If R is the fourth vertex of the rectangle POQR show that R parametrically is x=\frac{2at^2-4}{t},y=\frac{a(t^4+16)}{t^2} when st=-4

x=t-\frac{4}{t} y=\frac{t^2}{2}+\frac{8}{t^2} How can I solve that to get to x^2=2(y-4)

sorry its 1+cos2@

Thank you. And final question before I'm finished all this sh*t. Show\ that\ (1cos2\Theta +isin2\Theta )^2=2^ncos^n\Theta (cosn^\Theta +isin\ n\Theta )

I know this sound so simple, but I've never had to find the roots (apart for square) for something in cartesian form. I've always done it either just as a real number of just as an imaginary number. Find the five-fifth roots of \sqrt{3}+i and I'm tryna finish all my complex work tonight...

Thanks. I need help finishing off this question. I've found the five-fifth roots of uity and proved that if w=cis 2pi/5 1+w+w^2+w^3+w^4=0 But I cant show that w+w^4 and w^2+w^3 are the roots of z^2+z-1

I need help on this question. I did the first part and got |z|=2 as a locus But what is the locus of w where w=\frac{z-1}{z}

Thanks. I've got another one here. I've figured out that x^2+y^2=16 and x^2+y^2-24x-10y+88=0 touch each other externally. How do I show that 24x+10y-104=0 is a common tangent

Find the equation of the tangent at the point (3,-4) to the circle x^2+y^2=25. I got the answer to be 3x-4y=25 What are the equations of the two tangets parallel to the y-axis? I got x=5 and x=-5 Show that the first tanget intersects these tangets in points which subtend a right angle at...

Thanks.I got it now. Also. For the polynomial z^4 -2z^3 +7z^2 -4z +10, find real values of a for which ai is a solution. I know it has something to do with conjugate root theorem, but I'm not sure how to apply it without actual numbers.

I can't seem to se how the last line proves they're collinear.

Yeah. That's why I've never been able to implicitly differentiate anything when the constant is 0.

Wait. I thought you can't do implicit differentiation when the constant is 0 because when solving for vertical and horizontal tangents, it will keep coming out as 0.

The question was Re(z^2)+Im(z^2)=0

Thanks. One last thing. If I was to find the locus of z^2 where z=x+iy, what would the graph look like?

Gotcha. Haven't done conics that's why it's new to me. Also, how do you draw the ellipse with complex number |z-z1|+|z-z2|=2a

Yeah something like that where x^2-y^2=constant

How do you draw x^2-y^2

How do you draw arg(z-a)=arg(ia) At first I thought it was the 2 lines extendng from points a and ia, but then I realised arg(ia) isn't the same as arg(z-ia)

What's D?