If the chord $$y = mx + 1$$   of the circle $${x^2} + {y^2} = 1$$   subtends an angle of measure $${45^ \circ }$$ at the major segment of the circle then value of $$m$$ is-

Solution :
Equation of circle $${x^2} + {y^2} = 1 = {\left( 1 \right)^2}$$
$$\eqalign{ & \Rightarrow {x^2} + {y^2} = {\left( {y - mx} \right)^2} \cr & \Rightarrow {x^2} = {m^2}{x^2} - 2\,mxy \cr & \Rightarrow {x^2}\left( {1 - {m^2}} \right) + 2\,mxy = 0 \cr} $$
Which represents the pair of lines between which the angle is $${45^ \circ }$$
$$\eqalign{ & \tan \,{45^ \circ } = \pm \frac{{2\sqrt {{m^2} - 0} }}{{1 - {m^2}}} = \frac{{ \pm 2m}}{{1 - {m^2}}}; \cr & \Rightarrow 1 - {m^2} = \pm 2m \Rightarrow {m^2} \pm 2m - 1 = 0 \cr & \Rightarrow m = \frac{{ - 2 \pm \sqrt {4 + 4} }}{2} = \frac{{ - 2 \pm 2\sqrt 2 }}{2} = - 1 \pm \sqrt 2 \cr} $$