$$A)$$ Lines x−$$2y$$−$$6=0$$, $$3x+y$$−$$4=0$$ and λ$$x+4y+$$λ$$2=0$$ are concurrent. The value of λ $$isB)$$ The points $$($$λ$$+1$$,$$1)$$, (2$$λ$$+1$$,$$3) and (2$$λ$$+2$$,$$2$$λ$$) are collinear, then the value of λ $$isC)$$ If line $$x+y$$−$$1$$−λ$$=0$$, passing through the intersections of x $$-$$ y $$+$$ $$1$$ $$=$$ $$0$$ and $$3x+y$$−$$5=0$$ is perpendicular to one of them then the value of λ $$isD)$$ If the line y−x−$$1+$$λ$$=0$$ is equally inclined to axes and equidistant from the points (1$$, $$-2) and (3$$, $$4) then λ $$isp)$$ $$2q)$$ $$-4r)$$ $$-12s)$$ $$1$$

Question

$$A)$$ Lines x−$$2y$$−$$6=0$$, $$3x+y$$−$$4=0$$ and λ$$x+4y+$$λ$$2=0$$ are concurrent. The value of λ $$isB)$$ The points $$($$λ$$+1$$,$$1)$$, (2$$λ$$+1$$,$$3) and (2$$λ$$+2$$,$$2$$λ$$) are collinear, then the value of λ $$isC)$$ If line $$x+y$$−$$1$$−λ$$=0$$, passing through the intersections of x $$-$$ y $$+$$ $$1$$ $$=$$ $$0$$ and $$3x+y$$−$$5=0$$ is perpendicular to one of them then the value of λ $$isD)$$ If the line y−x−$$1+$$λ$$=0$$ is equally inclined to axes and equidistant from the points (1$$, $$-2) and (3$$, $$4) then λ $$isp)$$ $$2q)$$ $$-4r)$$ $$-12s)$$ $$1$$

Infinity Learn · Accepted Answer

$$A)$$ $$1$$−$$2$$−$$631$$−$$4$$λ$$4$$λ$$2=0$$⇒$$1($$λ$$2+16)+2(3$$λ$$2+4$$λ$$)$$−$$6(12$$−λ$$)=0$$⇒$$7$$λ$$2+14$$λ−$$56=0$$⇒λ$$2+2$$λ−$$8=0$$⇒λ$$=2$$,−$$4B)$$ Collinear ⇒$$2$$λ$$=2$$λ−$$3$$⇒λ$$=$$−$$12$$,$$2C)$$ Point of intersection of $$x-y+1=0$$ and $$3x+y-5=0$$ is (1$$,$$2). (1$$,$$2) lies $$on-x+y-1-$$λ$$=0$$⇒λ$$=2$$ $$D)$$ Perpendicular distance form (1$$,$$-2) and (3$$,$$4) to $$y-x-1$$ $$+$$λ$$=0$$ are equal⇒|λ−$$4$$|$$2=$$|λ|$$2$$⇒λ−$$4=$$−λ⇒λ$$=2$$

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