The equation of line of shortest distance between the lines x−$$31=y$$−$$5$$−$$2=z$$−$$71$$;$$x+17=y+1$$−$$6=z+11$$ is

Question

The equation of line of shortest  distance between the lines x−$$31=y$$−$$5$$−$$2=z$$−$$71$$;$$x+17=y+1$$−$$6=z+11$$   is

Infinity Learn · Accepted Answer

Let  $$P($$α$$+3$$, −$$2$$α$$+5$$, α$$+7)$$ and $$Q(7$$β−$$1$$, −$$6$$β−$$1$$, β−$$1)$$ be the points on the given lines so that PQ is the line of shortest distance between the given linesD.r’s of $$PQ=($$α−$$7$$β$$+4$$, −$$2$$α$$+6$$β$$+6$$, α−β$$+8)Since$$ PQ is perpendicular to the given lines $$1($$α−$$7$$β$$+4)$$−$$2($$−$$2$$α$$+6$$β$$+6)+1($$α−β$$+8)=0$$⇒$$6$$α−$$20$$β$$=0$$⇒$$3$$α−$$10$$β$$=0$$ $$---$$ (1)and$$,⇒$$7($$α−$$7$$β$$+4)−$$6($$−$$2$$α$$+6$$β$$+6)+1($$α−β$$+8)=0$$⇒$$7$$α−$$49$$β$$+28+12$$α−$$36$$β−$$36+$$α−β$$+8=0$$ ⇒$$20$$α−$$86$$β$$=0$$⇒$$10$$α$$=43$$β$$---$$ (2)From$$ $$(1) and (2) α$$=0$$, β$$=0P=(3$$, $$5$$, $$7)$$, $$Q=($$−$$1$$, −$$1$$, −$$1)D$$.r’s of PQ $$=$$ $$($$–$$4$$, –$$6$$, –$$8)$$ $$=$$ $$(2$$, $$3$$, $$4)Equation$$ of PQ is x−$$32=y$$−$$53=z$$−$$74$$

The equation of line of shortest distance between the lines x−31=y−5−2=z−71;x+17=y+1−6=z+11 is

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