Let 1 2 be two parametric curves 1 z t i0 0 t 2 and 2 z

Let 1, 2 be two parametric curves 1 : z = t + i0, 0 t 2 and 2 : z = + i ( 1), 0 2. Let 1, 2 be their images under the mapping w = iz2 1. Find the points z1 and z2 at which 1 and 2 intersect and determine the corresponding angles of intersection, 1 and 2. Sketch 1 and 2. 2. Find the parametric equations for 1 and 2. 3. Find the points w1 and w2 at which 1 and 2 intersect and determine the corresponding angles of intersection, 1 and 2. Sketch 1 and 2

Solution

they will intersect when real part and imaginary part of both y1 and y2 are equal

it means t=

0= ( 1)

it means =0,1 and t=0,1

z1=0,z2=1

to find angle of intersection

find value of slope at point of intersections- differentiate the functions and put the value of t or .

at z1=0, slope of y1,m1=1 slope of y2,m2=1+i(-1)

angle between it can be found by cos=m1.m2/(lm1llm2l)

cos1=1/(2)^0.5 1=45 degree

similarly 2 can be found out.

plot= y1 is x axis. y2 y=x(x-1) simple quadratic, plot it.

now to find the image

w=i(x+iy)^2=i(x^2+-y^2+2xyi)=-2xy+i(x^2-y^2)

put x=t,y=0

X+iY=0+i(t^2)

so w1= it^2

similarly repeat for w2 .

now just repeat same for w1,w2 what i did with y1 and y2.