Complex plane locus question (another one)

applestrudle
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Homework Statement



again there is no answer provided in the book!

a +bt +ct^2 = z where t is a real parameter, and a, b, c are complex numbers with b/c real


Homework Equations



The Attempt at a Solution



b/c real indicates that b and c are pure imaginary so when you split the equation into real and imaginary parts:

x = Re(a)

y = Im(a) + Im(b)t + Im(c)t^2

since t is a parameter, in the complex plane, the locus of z is a straight vertical line with x = Re(a)?
 
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applestrudle said:
b/c real indicates that b and c are pure imaginary

Are you sure? What if b = 1 + i and c = 2 + 2i ?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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