Parabola Part 1

Given, parabola
y2=8x...(i)
Equation of tangent at P(2,−4) is
−4y=4(x+2)
orx+y+2=0 ...(ii)
and Equation of normal to the parabola is
x−y+C=0
∴ Normal passes through (2,−4)
∴C=−6
Normal:x−y=6 ...(iii)
Equation of directrix of parabola
x = − 2 ...(iv)
Point of intersection of tangent and normal with directrix are x = −2 at A(−2,0) and B(−2,−8) respectively.
Q(a,b) and P(2,−4) are given and AQBP is a square.
Mid-point of AB = Mid-point of PQ
⇒(−2,−4)=(

a+2

2

,

b−4

2

)
⇒a=−6,b=−4
⇒2a+b=−16