If one of the roots of the equation $x^2-5 x-14=0$ is the length of the semi conjugate axis of the hyperbola…
If one of the roots of the equation $x^2-5 x-14=0$ is the length of the semi conjugate axis of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and the square of the other root is the semi-transverse axis then the focus of the hyperbola that lies on the positive $x$-axis is
$(5,0)$
$(\sqrt{65}, 0)$
$(7,0)$
$(\sqrt{74}, 0)$
Solution
Given equation is $x^2-5 x-14=0$
how roots are $-2,7$
Let $b=7$ and $a=4$
hence $C=\sqrt{7^2+9^2}=\sqrt{65}$
hence $S=(\sqrt{65}, 0)$