The angle between the tangents drawn from the· origin to the circle  $(x-7{)}^2+(y+1{)}^2=25,$ is

Let y = mx be a tangent drawn from the origin to 

the circle  $e(x-7{)}^2+(y+1{)}^2=5^2$ then

  $\frac{7m-(-1)}{\sqrt{m^2}+1}=\pm 5\Rightarrow 12m^2+7m-12=0$

Let  $m_1$ and $m_2$ be the slopes of the two tangents. Then,

  $m_1{m}_2=-\frac{12}{12}=-1$ 

Hence, the angle between two tangents is  $\pi /2$ 

AUTER The director circle of the given circle is 

$(x-7{)}^2+(y+1{)}^2=50$

 Clearly,  $( 0, 0)$  lies on this circle. So, required angle is  $\pi /2$