The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x = \pm \frac { 4 } { \sqrt { 13 } }$, respectively. Let the line $y - \sqrt { 3 } x + \sqrt { 3 } = 0$ touch this hyperbola at $( x _ { 0 } , y _ { 0 } )$. If m is the product of the focal distances of the point $\left( x _ { 0 } , y _ { 0 } \right)$, then $4 \mathrm { e } ^ { 2 } + \mathrm { m }$ is equal to $\_\_\_\_$

Q83. The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x = \pm \frac { 4 } { \sqrt { 13 } }$, respectively. Let the line $y - \sqrt { 3 } x + \sqrt { 3 } = 0$ touch this hyperbola at ( $x _ { 0 } , y _ { 0 }$ ). If m is the product of the focal distances of the point $\left( x _ { 0 } , y _ { 0 } \right)$, then $4 \mathrm { e } ^ { 2 } + \mathrm { m }$ is equal to $\_\_\_\_$

Q67. Let $H : \frac { - x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ be the hyperbola, whose eccentricity is $\sqrt { 3 }$ and the length of the latus rectum is $4 \sqrt { 3 }$. Suppose the point $( \alpha , 6 ) , \alpha > 0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $( \alpha , 6 )$, then $\alpha ^ { 2 } + \beta$ is equal to
(1) 172
(2) 171
(3) 169
(4) 170