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Question

# $A,B,C,D$ are four points on a plane with position vectors $\text{a,b,candd}$ respectively such that $\left(a-d\right).\left(b-c\right)=\left(b-d\right).\left(c-d\right)=0$. For $△ABC,D$ is the?

A

Incentre

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B

orthocentre

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C

centroid

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D

None of these

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Solution

## The correct option is B orthocentreExplanation for the correct option:Using the concept of dot product:According to the question,$\because \left(a-d\right).\left(b-c\right)=0$$⇒$ $DA.CB=0$ $\therefore AD\perp BC$ $\left[\because \stackrel{\to }{a}.\stackrel{\to }{b}=0⇒\stackrel{\to }{a}\text{isperpendicularto}\stackrel{\to }{b}\right]$Again, $\because \left(b-d\right).\left(c-a\right)=0$$⇒$ $DB.AC=0$$\therefore BD\perp CA$Thus, $D$ is the intersection of the altitudes passing through $A$ and $B$.$\therefore D$ is the orthocentre of $△ABC$.Hence, option ‘B’ is correct.

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