The vertices of a triangle are
`[at_1t_2, a(t_1+t_2)],[at_2t_3,a(t_2+t_3)],[at_3t_1,a(t_3+t_1)]`
Find the orthocentre of the triangle.

The vertices of a triangle are [a t_1t_2,a(t_1 +t_2)] , [a t_2t_3,a(t_2 +t_3)] , [a t_3t_1,a(t_3 +t_1)] Then the orthocenter of the triangle is (a) (-a, a(t_1+t_2+t_3)-at_1t_2t_3) (b) (-a, a(t_1+t_2+t_3)+at_1t_2t_3) (c) (a, a(t_1+t_2+t_3)+at_1t_2t_3) (d) (a, a(t_1+t_2+t_3)-at_1t_2t_3)

The coordinates of the vertices of a triangle are [at_(1)t_(2),a(t_(1)+t_(2))],[at_(2),t_(3),a(t_(2)+t_(3))] and [at_(3)t_(1),t_(3)+t_(1))] . Find the coordinates of the ortocentre of the triangle.

If the Verticesof a triangle are P(at_1,a/t_1) , Q(at_2,a/t_2) and R(at_3,a/t_3) Show that the orthocentre of the triagle lies on xy = a^2 .

If t_1,t_2 and t_3 are distinct, the points (t_1, 2at_1 + at_1^3), (t_2, 2at_2 + at_2^3), (t_3, 2at_3 +at_3^3) are collinear then show that t_1+t_2+t_3=0 .

Prove that the area of the triangle whose vertices are (t ,t-2),(t+2,t+2), and (t+3,t) is independent of tdot

If t_1, t_2 and t_3 are distinct, the points (t_1, 2at_1+at_1^3), (t_2,2at_2+at_2^3) and (t_3,2at_3+at_3^3) are collinear if

The base of a triangle is divided into three equal parts. If t_1, t_2,t_3 are the tangents of the angles subtended by these parts at the opposite vertex, prove that (1/(t_1)+1/(t_2))(1/(t_2)+1/(t_3))=4(1+1/(t2 2))dot

Let A(ct_(1), (c)/(t_(1))), B(ct_(2),(c)/(t_(2))) and C(ct_(3),(c)/(t_(3))) be the vertices of the triangle ABC. Show that the ortocentre of the triangle lies on xy=c^(2) .

Show that the relation R defined in the set A of all triangles as R={(T _(1), T _(2))):T _(1) is similar to T _(2) } is equivalence relation. Consider three right angle triangles T_(1) with sides 3,4,5,T_(2) with sides 5,12,13 and T_(3) with sides 6,8,10. Which triangles among T_(1), T_(2) and T_(3) are related ?

Answer the following questions : (Alternatives are to be noted): If T_1 , T_2 , T_3 be estimators with expectations E(T_1) = theta_1+ 2theta_2 , E(T_2) = theta_2+2theta_3 , and E(T_3) = theta_3 + 2theta_1 find unbaised estimator of theta_1+theta_2+theta_3 .