Find a parametrization of the intersection of the surfaces z

Find a parametrization of the intersection of the surfaces z = x2 y2 and z = x2 + xy 5 using t = y as a parameter.

Solution

z= x^2 - y^2

z = x^2 +xy -5

use y =t

point of intersection of surfaces : x^2 -y^2 = x^2 +xy -5

-y^2 = xy -5

y^2 +xy -5 =0

plug t = y

So, t^2 +xt -5 =0

xt = 5 - t^2

x = 5/t - t

z = x^2 - y^2 = (5/t -t)^2 - t^2

= 25/t^2 +t^2 - 10 -t^2

= 25/t^2 -10

So, x = 5/t -t

y = t

z = 25/t^2 -10

So, point of intersection : ( 5/t -t , t)