Question

Calculate the area for the trapeze `PQRS` ? (See question)

Answer 1

`15` area units.

Explanation:

Given two conics

`C_1(x_1,y_1)=x_1^2+y_1^2-2=0` and
`C_2(x_2,y_2)=y_2^2-8x_2=0`

The tangent space to those conics is

`m_1 = (dy_1)/(dx_1) = -x_1/y_1`
`m_2 = (dy_2)/(dx_2) = 4/y_2`

the tangency condition reads

`{(y_2-y_1=m_1(x_2-x_1)),(y_1-y_2=m_2(x_1-x_2)):}`

Solving now

`{(x_1^2+y_1^2-2=0),(y_2^2-8x_2=0),(y_2-y_1=m_1(x_2-x_1)),(y_1-y_2=m_2(x_1-x_2)):}`

for `x_1,x_2,y_1,y_2` we obtain

`((Q,(-1,-1)),(P,(-1,1)),(S,(2,-4)),(R,(2,4)))`

Attached a plot focusing the main elements

The area is `(3(2+8))/2=15`.