Question

Two systems of rectangular axes have the same origin

Answer 1

Perpendicular diatance of the plane ax+by+cz+d=0ax+by+cz+d=0from origin is ∣∣∣da2+b2+c2−−−−−−−−√∣∣|da2+b2+c2|If two system of lines have the same origin then their ⊥⊥distance from origin to the plane in both the system are equal.

Let the equation of the plane in both the systems be

xa+yb+zcxa+yb+zc=1=1 and Xa′+Yb′+Zc′Xa′+Yb′+Zc′=1=1

We know that ⊥⊥ diatance of the plane ax+by+cz+d=0ax+by+cz+d=0 from origin is ∣∣∣da2+b2+c2−−−−−−−−√∣∣|da2+b2+c2|

It is given that the origin is same for both the system.

Hence ⊥⊥ distance of the plane from origin are equal for both the system.

(ie)∣∣∣−11a2+1b2+1c2−−−−−−−−−−√∣∣(ie)|−11a2+1b2+1c2|=∣∣∣−11a′2+1b′2+1c′2−−−−−−−−−−−√∣∣=|−11a′2+1b′2+1c′2|

⇒1a2+1b2+1c2=1a′2+1b′2+1c′2⇒1a2+1b2+1c2=1a′2+1b′2+1c′2

Hence proved.

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