question_answer

If two vertices of a triangle are (5, -1) and (-2,3) and its orthocentre is at (0, 0), then the third vertex is                JEE Main Online Paper (Held On 12 May 2012)             

A) (4,-7)                                    

B)                        (-4,-7)

C)                        (-4,7)                                    

D)                        (4,7)

Correct Answer: B

Solution :

                              Let the third vertex of \[\Delta ABC\] be (a,b). Ortho centre =H (0,0) Let A (5, - 1) and B (- 2, 3) be other two vertices of \[\Delta ABC.\] Now, (Slope of AH) \[\times \](Slope of BQ =-1 \[\Rightarrow \]\[\left( \frac{-1-0}{5-0} \right)\left( \frac{b-3}{a+2} \right)=-1\]                 ...(1) \[\Rightarrow \]\[b-3=5(a+2)\]Similarly, (Slope of BH) \[\times \] (Slope of AC) =-1 \[\Rightarrow \]\[-\left( \frac{3}{2} \right)\times \left( \frac{b+1}{a-5} \right)=-1\] \[\Rightarrow \]\[3b+3=2a-10\]\[\Rightarrow \]\[3b-2a+13=0\]?(2) On solving equation (1) and (2) we get \[a=-4,b=-7\] Hence, third vertex is (-4,-7).