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MCQMediumJEE 2023Basics of Vectors

JEE Mathematics 2023 Question with Solution

If the points P and Q are respectively the circumcenter and the orthocenter of a △ABC\triangle ABC, the PA→+PB→+PC→\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC} is equal to:

  • A

    2PQ→2 \overrightarrow{PQ}

  • B

    PQ→\overrightarrow{PQ}

  • C

    2PQ→2 \overrightarrow{PQ}

  • D

    PQ→\overrightarrow{PQ}

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: P is the circumcenter and Q is the orthocenter of △ABC\triangle ABC.

Find: The value of PA→+PB→+PC→\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC}.

Let the position vectors of A,B,CA, B, C be a→,b→,c→\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} respectively.

From the relation used in the solution,

PA→+PB→+PC→=a→+b→+c→\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC} = \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}

The centroid GG satisfies

PG→=a→+b→+c→3\overrightarrow{PG} = \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3}

So,

a→+b→+c→=3PG→\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 3\overrightarrow{PG}

Hence,

PA→+PB→+PC→=3PG→=PQ→\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC} = 3\overrightarrow{PG} = \overrightarrow{PQ}

Therefore, the required vector is PQ→\overrightarrow{PQ}.

The solution explicitly marks option C as correct, although the displayed option texts contain duplicate entries and the concluding line incorrectly says option (3) while the answer key says (4). Since the solution is the primary source, the correct option is taken as C.

Consistency Check

The extracted page has an internal inconsistency:

  • The header says The Correct Option is C.
  • The final sentence says correct option (3).
  • The answer key says (4).
  • Options A and C are identical, and options B and D are identical.

Because the solution is the primary source, the answer is resolved as C. The underlying vector result stated in the working is PQ→\overrightarrow{PQ}.

Common mistakes

  • Assuming the answer must be a scalar multiple of a centroid vector without using the given relation between circumcenter, centroid, and orthocenter. This misses the triangle center geometry. Use the standard relation connecting these centers before comparing vectors.

  • Choosing between the duplicated options only from the answer key. Here the option texts repeat, so the page working must be used as the primary provided. Always resolve such conflicts from the solution first.

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