JEE Main & Advanced Mathematics Three Dimensional Geometry Equation of Planes Bisecting Angle Between Two Given Planes

Equations of planes bisecting angles between the planes \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0\] and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+d=0\] are \[\frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{(a_{1}^{2}+b_{1}^{2}+c_{1}^{2})}}=\] \[\pm \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{(a_{2}^{2}+b_{2}^{2}+c_{2}^{2})}}\].

(i) If angle between bisector plane and one of the plane is less than \[{{45}^{o}}\], then it is acute angle bisector, otherwise it is obtuse angle bisector.

(ii) If \[{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}\] is negative, then origin lies in the acute angle between the given planes provided \[{{d}_{1}}\] and \[{{d}_{2}}\] are of same sign and if \[{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}\] is positive, then origin lies in the obtuse angle between the given planes.