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QJEEM16Ma3878: JEE Main 2016 Physics - Class 11 JEEM16Ma3878 Solution: Let ABC be a triangle whose circumcentre is at P. If the…

Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are a→,b→,c→\overrightarrow a ,\overrightarrow b ,\overrightarrow c and a→+b→+c→4{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4} respectively, then the position vector of the orthocentre of this triangle, is :

Multiple Choice Options

  • A a→+b→+c→{\overrightarrow a + \overrightarrow b + \overrightarrow c }
  • B −(a→+b→+c→2)- \left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}} \right)
  • C 0→\overrightarrow 0
  • D (a→+b→+c→2)\left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}} \right)

Step-by-Step Verified Solution

JEE Main 2016 (Online) 10th April Morning Slot Mathematics - Vector Algebra Question 197 English Explanation

Given,

Position vector of circumcentre, C→=a→+b→+c→4\overrightarrow C = {{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}

We know, position vector of centroid, G→=a→+b→+c→3\overrightarrow G = {{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 3}

Now, let R→\overrightarrow R be the orthocentre of the triangle.

We know, G→\overrightarrow G =2C→+R→3= {{2\overrightarrow C + \overrightarrow R } \over 3}

⇒\Rightarrow    3 G→\overrightarrow G =2C→+R→= 2\overrightarrow C + \overrightarrow R

⇒\Rightarrow     R→=3G→−2C→\overrightarrow R = 3\overrightarrow G - 2\overrightarrow C

=    (a→+b→+c→)−2(a→+b→+c→4)\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) - 2\left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}} \right)

=    a→+b→+c→2{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}

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