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  • The vertices of a triangle are <img alt="\mathrm{ \left[a t_1 t_2, a\left(t_1+t_2\right)\right],\left[a t_2 t_3, a\left(t_2+t_3\right)\right],\left[a t_3 t_1, a\left(t_3+t_1\right)\right] .

The vertices of a triangle are
\mathrm{ \left[a t_1 t_2, a\left(t_1+t_2\right)\right],\left[a t_2 t_3, a\left(t_2+t_3\right)\right],\left[a t_3 t_1, a\left(t_3+t_1\right)\right] . }
Find the abscissa of orthocentre of the triangle.
 

Option: 1

\mathrm{a t_1 t_2 t_3}
 


Option: 2

\mathrm{a\left(t_1+t_2+t_3\right)}
 


Option: 3

\mathrm{-a}
 


Option: 4

\mathrm{a}


Answers (1)

best_answer

Let \mathrm{A, B, C} be the vertices

\mathrm{ A \equiv\left[a t_1 t_2 a\left(t_1+t_2\right)\right] }

\mathrm{ B \equiv\left[a t_2 t_3, a\left(t_2+t_3\right)\right] }

\mathrm{ C \equiv\left[a t_3 t_1, a\left(t_3+t_1\right)\right] }

Slope of \mathrm{ B C=\frac{a\left(t_3+t_7\right)-a\left(t_2+t_3\right)}{a t_3 t_I-a t_2 t_3}=\frac{l}{t_3} }

Slope of altitude \mathrm{A D=-t_3}

Equation of \mathrm{A D} is

\mathrm{ y-a\left(t_1+t_2\right)=-t_3\left(x-a t_1 t_2\right) }           (1)

Similarly equation of altitude \mathrm{ B E } is

\mathrm{ y-a\left(t_2+t_3\right)=-t_1\left(x-a t_2 t_3\right) }

The coordinates of orthocentre \mathrm{ O } are obtained by simultaneously solving equation (1) and equation (2).

Subtracting equation (2) from (1)

\mathrm{ a\left(t_3-t_1\right)=-x\left(t_3-t_1\right) \Rightarrow x=-a }

Hence option 3 is correct.
 

Posted by

Gaurav

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