Let P be a point in the first octant, whose image Q in the plane $$x+y=3$$ (that$$ is, the line segment PQ is perpendicular to the plane $$x+y=3$$ and the $$mid-point$$ of PQ lies in the plane $$x+y=3$$ $$) lies on the $$z-axis$$. Let the distance of P from the $$x-axis$$ be $$5$$. If R is the image of P in the $$xy-plane$$, then the length of PR is

Question

Let P be a point in the first octant, whose image Q in the plane $$x+y=3$$ (that$$ is, the line segment PQ is perpendicular to the plane $$x+y=3$$  and the $$mid-point$$ of PQ lies in the plane $$x+y=3$$ $$) lies on the $$z-axis$$. Let the distance of P from the $$x-axis$$ be $$5$$. If R is the image of P in the $$xy-plane$$, then the length of PR is

Infinity Learn · Accepted Answer

Let $$P($$α,β,γ$$)$$ ,since Q lies on Z axis,let Q $$be(0$$,$$0$$,γ$$)$$ and R is image ofP in the XY plane,so let R $$be=($$α,β,$$-$$γ$$)$$ Now, PQ¯‖$$i^+j^$$⇒$$($$α$$i^+$$β$$j^)$$‖(i^+j^)⇒α$$=$$β  Also, mid point of PQ $$=$$α$$2$$,β$$2$$,γ lies on the plane $$x+y=3$$⇒α$$2+$$β$$2=3$$⇒α$$+$$β$$=6$$⇒α$$=3$$ Now, distance of point P from X $$-axis$$ $$=$$β$$2+$$γ$$2=5$$⇒β$$2+$$γ$$2=25$$⇒γ$$2=16$$  as β$$=$$α$$=3$$  as γ$$=4$$  Hence $$PR=2$$γ$$=8$$

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