question_answer

The orthocenter of the triangle with vertices \[A(0,0),\] \[B\left( 0,\frac{3}{2} \right),\] \[C(-5,0)\]is

A)  \[\left( \frac{5}{2},\frac{3}{4} \right)\]                  

B)  \[\left( \frac{-5}{2},\frac{3}{4} \right)\]

C)  \[\left( -5,\frac{3}{2} \right)\]                   

D)  \[(0,0)\]

Correct Answer: D

Solution :

Let, \[\Delta AOB\] is the given triangle Slope of  \[AB=\frac{\frac{3}{2}-0}{0+5}=\frac{3}{10}\] Slope of  \[BO=\frac{0-0}{0+5}=0\] The equation of line passing through A and perpendicular to BO is  \[y-0=-0\left( x-\frac{3}{2} \right)\]                                 \[\Rightarrow \]                               \[y=0\]                 ?..(i) and equation of line passing through 0 and perpendicular to AB is \[y-0=-\frac{10}{3}(x-0)\] \[\Rightarrow \]                               \[y=-\frac{10}{3}x\]        ?..(ii) The intersection point of Eqs. (i) and (ii) is \[(0,0)\] which is the required orthocentre.