A $$)$$ If a, b, c are in A.P, then lines $$ax+by+c=0$$ are $$concurrentB)$$ A point on the line x $$+$$ y $$=$$ $$4$$ which lies at a unit distance from the line $$4x$$ $$+$$ $$3y$$ $$=$$ $$10$$ $$isC)$$ Orthocentre of triangle made by lines x $$+$$ y $$=$$ $$1$$, $$x-$$ y $$+$$ $$3$$ $$=$$ $$0$$, $$2x$$ $$+$$ y $$=$$ $$7$$ $$isD)$$ Two vertices of a triangle are (5$$, $$-1) and $$(-2$$,$$3)$$. If orthocentre is the origin then coordinate of the third vertex $$arep)$$ $$-4$$,$$-7q)$$ $$-7$$,$$11r)$$ $$1$$,$$-2s)$$ $$-1$$,$$2t)$$ $$3$$,$$1$$

Question

A $$)$$ If a, b, c are in A.P, then lines $$ax+by+c=0$$ are $$concurrentB)$$  A point on the line x $$+$$ y $$=$$ $$4$$ which lies at a  unit distance from the line $$4x$$ $$+$$ $$3y$$ $$=$$ $$10$$ $$isC)$$ Orthocentre of triangle made by lines x $$+$$ y $$=$$ $$1$$, $$x-$$ y $$+$$ $$3$$ $$=$$ $$0$$, $$2x$$ $$+$$ y $$=$$ $$7$$ $$isD)$$ Two vertices of a triangle are (5$$, $$-1) and $$(-2$$,$$3)$$. If orthocentre is the origin then coordinate of the third vertex $$arep)$$ $$-4$$,$$-7q)$$ $$-7$$,$$11r)$$ $$1$$,$$-2s)$$ $$-1$$,$$2t)$$ $$3$$,$$1$$

Infinity Learn · Accepted Answer

$$A)$$ $$a+c=2b$$⇒$$a+c$$−$$2b=0$$ ⇒$$ax+by+c=0Satisfies$$ (1$$, $$-2)B) Perpendicular distance of $$P($$λ,$$4$$−λ$$)$$ from $$4x+3y=10$$|$$4$$λ$$+3(4$$−λ$$)$$−$$10$$|$$5=1$$⇒|λ$$+2$$|$$=5$$  ⇒λ−$$2$$±$$5=3$$,−$$7P(3$$,$$1)$$, $$($$−$$7$$,$$+11)C)$$ Point $$B=($$−$$1$$,$$2)D)$$ $$yx=744+1y$$−$$5=23$$,$$214x=2x$$−$$13x=$$−$$4$$,$$y=$$−$$7$$

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