A tangent and a normal are drawn at the point P(2, $-$4) on the parabola y² = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to :

<p>Given, parabola</p> <p>${y^2} = 8x$ ...... (i)</p> <p>Equation of tangent at $P(2, - 4)$ is</p> <p>$- 4y = 4(x + 2)$</p> <p>or, $x + y + 2 = 0$ ..... (ii)</p> <p>and Equation of normal to the parabola is</p> <p>$x - y + C = 0$</p> <p>$\therefore$ Normal passes through $(2, - 4)$</p> <p>$\therefore$ $C = - 6$</p> <p>Normal : $x - y = 6$ ..... (iii)</p> <p>Equation of directrix of parabola</p> <p>$x = - 2$ ..... (iv)</p> <p>Point of intersection of tangent and normal with directrix are $x = - 2$ at $A( - 2,0)$ and $B( - 2, - 8)$ respectively.</p> <p>$Q(a,b)$ and $P(2, - 4)$ are given and AQBP is a square.</p> <p>Mid-point of AB = Mid-point of PQ</p> <p>$$ \Rightarrow ( - 2, - 4) = \left( {{{a + 2} \over 2},{{b - 4} \over 2}} \right) \Rightarrow a = - 6,b = - 4$$</p> <p>$\Rightarrow 2a + b = - 16$</p>