If $y = x + 1$ is axis of parabola, $x + y = 4$ is tangent of same parabola at its vertex and $y = 2 x + 3$ is one of its tangents, then the length of the latus rectum of parabola is $\frac{a \sqrt{2}}{b}$ where a and b are relatively prime natural numbers, then $\frac{a + b + 1}{12} =$

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Detailed Solution

$O = (\frac{3}{2}, \frac{5}{2}); Q = (\frac{1}{3}, \frac{11}{3})$
feet of perpendicular from focus on any tangent lies on tangent at vertex (point Q)
Focus S = $(\frac{17}{9}, \frac{26}{9})$
Foot of directrix = $(\frac{10}{9}, \frac{19}{9})$
Eq.of directrix = $x + y = \frac{29}{9} = 9 x + 9 y - 29 = 0$
L.L.R = $4 \sqrt{{(\frac{17}{9} - \frac{3}{2})}^{2} + {(\frac{19}{9} - \frac{5}{2})}^{2}} = \frac{14 \sqrt{2}}{9}$