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Condition for Concurrency of Three Lines

The point of ...

Question

The point of concurrency of the altitudes drawn from the vertices $A (a t_{1} t_{2}, a (t_{1} + t_{2})), B (a t_{2} t_{3}, a (t_{2} + t_{3}))$ and $C (a t_{3} t_{1}, a (t_{3} + t_{1}))$ of the triangle $A B C$ (where $t_{1} \neq t_{2} \neq t_{3})$ is

(- a t_{1} t_{2} t_{3}, a (t_{1} + t_{2} + t_{3}))

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(- a, a (t_{1} + t_{2} + t_{3} + t_{1} t_{2} t_{3}))

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(- a t_{1} t_{2} t_{3}, a (t_{1} t_{2} + t_{2} t_{3} + t_{3} t_{1}))

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(0, 0)

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Solution

The correct option is B $(- a, a (t_{1} + t_{2} + t_{3} + t_{1} t_{2} t_{3}))$

$m_{A B} = \frac{1}{t_{2}}, m_{B C} = \frac{1}{t_{3}}$ and $m_{C A} = \frac{1}{t_{1}}$
$∵ A D ⊥ B C, C F ⊥ A B$ and $B E ⊥ A C$
$∴ m_{A D} = - t_{3}, m_{C F} = - t_{2}$ and $m_{B E} = - t_{1}$

Equation of $A D$ is $y - a (t_{1} + t_{2}) = - t_{3} (x - a t_{1} t_{2}) \dots (1)$
Equation of $C F$ is $y - a (t_{3} + t_{1}) = - t_{2} (x - a t_{3} t_{1}) \dots (2)$

Solving equations $(1)$ from $(2)$ , we get
$x = - a$ and $y = a (t_{1} + t_{2} + t_{3} + t_{1} t_{2} t_{3})$

Hence, the point of concurrency of the altitudes is
$(- a, a (t_{1} + t_{2} + t_{3} + t_{1} t_{2} t_{3}))$ .

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Condition for Concurrency of Three Lines

Standard XII Mathematics

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The point of concurrency of the altitudes drawn from the vertices A(at1t2,a(t1+t2)),B(at2t3,a(t2+t3)) and C(at3t1,a(t3+t1)) of the triangle ABC (where t1≠t2≠t3)is

The point of concurrency of the altitudes drawn from the vertices $A (a t_{1} t_{2}, a (t_{1} + t_{2})), B (a t_{2} t_{3}, a (t_{2} + t_{3}))$ and $C (a t_{3} t_{1}, a (t_{3} + t_{1}))$ of the triangle $A B C$ (where $t_{1} \neq t_{2} \neq t_{3})$ is