Question

The vertices of a triangle are $\lbrack at_{1}t_{2},a(t_{1} + t_{2})\rbrack,\lbrack at_{2}t_{3},a(t_{2} + t_{3})\rbrack$, $\lbrack at_{3}t_{1},a(t_{3} + t_{1})\rbrack$, then the coordinates of its orthocentre are.

• A. $\lbrack a,a(t_{1} + t_{2} + t_{3} + t_{1}t_{2}t_{3})\rbrack$
• B. $\lbrack - a,a(t_{1} + t_{2} + t_{3} + t_{1}t_{2}t_{3})\rbrack$
• C. $\lbrack - a(t_{1} + t_{2} + t_{3} + t_{1}t_{2}t_{3}),a\rbrack$
• D. None of these

Answer 1

Answer: (B)

$\lbrack - a,a(t_{1} + t_{2} + t_{3} + t_{1}t_{2}t_{3})\rbrack$

Answer 2

$m _ { 1 } = \frac { - a \left( t _ { 1 } t _ { 2 } - t _ { 2 } t _ { 3 } \right) } { a \left( t _ { 1 } - t _ { 3 } \right) } = - t _ { 2 } , m _ { 2 } = - t _ { 3 }$

Therefore, perpendicular from point third

$t _ { 2 } x + y = a \left[ t _ { 1 } t _ { 2 } t _ { 3 } + t _ { 1 } + t _ { 3 } \right]$and perpendicular from point first $t _ { 3 } x + y = a \left[ t _ { 1 } t _ { 2 } t _ { 3 } + t _ { 1 } + t _ { 2 } \right]$

$x = - a , y = a \left[ t _ { 1 } t _ { 2 } t _ { 3 } + t _ { 1 } + t _ { 2 } + t _ { 3 } \right]$.