Let (-1, -1) be the focus of a parabola and 3x-y=8 be the tangent drawn to it at point (7,13) on the parabola. If ax + by + 19=0 is the equation of directrix of the parabola then the value of a+b is (where a, b $\in$ N)

Let foot of perpendicular from $\mathrm{S}(-1, -1) \mathrm{on} 3\mathrm{x}-\mathrm{y}=8 \mathrm{is} (\mathrm{\alpha },\mathrm{\beta })$

$\therefore \frac{\mathrm{\alpha }+1}{3}=\frac{\mathrm{\beta }+1}{-1}=1 \therefore (\mathrm{\alpha },\mathrm{\beta })=(-2,-2)$

Image of S about the tangent $3\mathrm{x}-\mathrm{y}=8$S’(5, -3) which is on directrix

Let point of intersection of tangent and directrix =$Q\left(h,3h-8\right)$

P(7,13) is poiint of contact of tangent

$$\begin{gathered} \angle QSP=90°\Rightarrow \left(slope of QS\right)\left(Slope of SP\right)=-1 \\ \Rightarrow \left(\frac{3h-7}{h+1}\right)\frac{7}{4}=-1\Rightarrow h=\frac{9}{5}\Rightarrow Q=\left(\frac{9}{5},\frac{-13}{5}\right) \\ Slope of directrix=\frac{\left(-13/5\right)+3}{9/5-5}=\frac{-1}{8} \\ \Rightarrow Equation of directrix \\ y+3=\frac{-1}{8}\left(x-5\right)\Rightarrow x+8y+19=0 \end{gathered}$$

$$\begin{gathered} \therefore {\mathrm{Eq}}^{\mathrm{n}} \mathrm{of} \mathrm{directrix} \mathrm{is} 8\mathrm{y}+\mathrm{x}+19=0 \\ \Rightarrow \mathrm{a}+\mathrm{b}=9 \end{gathered}$$