If Two Vertices Of A Triangle Are (–2, 3) And (5, –1), Orthocentre Lies At The Origin And Centroid On The Line x + y = 7, Then The Third Vertex Lies At

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Question

If two vertices of a triangle are (–2, 3) and (5, –1), orthocentre lies at the origin and centroid on the line x + y = 7, then the third vertex lies at

Solution

Correct option is

None of these

Let O(0, 0) be the orthocentre; A(h, k) the third vertex; B(–2, 3) and C(5,–1) the other two vertices.

Then the slope of the line through A and O is k/h, while the line throughB and C has the slope (–1 –3)/(5 + 2) = –4/7. By the property of the orthocentre, these two lines must be perpendicular, so we have   

             

 Also 

  

Which is not satisfied by the points given in options.

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