Prove that area of the trianlge whose vertices are `(at_1^2,2at_1) , (at_2^2,2at_2),(at_3^2,2at_3) ` is `a^2(t_1-t_2)(t_2-t_3)(t_3-t_1)`

Find the area of the triangle whose vertices are A(at_!^2, 2at_1),B(at_2^2, 2at_2) and C(at_3^2, 2at_3)

Show that the points (at_(1)^(2),2at_(1)) , (at_(2)^(2),2at_(2)) and (a,0) are collinear if t_1t_2 =-1 .

Statement I: The lines from the vertex to the two extremities of a focal chord of the parabola y^2=4ax are perpendicular to each other. Statement II: If the extremities of focal chord of a parabola are (at_1^2,2at_1) and (at_2^2,2at_2) , then t_1t_2=-1 .

The velocity-time graph of particle in one dimensional motion is shown in the fig. Which if the following formulae are correct for describing the motion of the pawrticle over the time interval t_1 to t_2 v(t_2) = v(t_1) + a(t_2 - t_2) , v_(average) = ([x(t_2) - x(t_1)])/((t_2 - t_1)) , x(t_2) - x(t_1) = area under the v - t curve. bounded by the t-axis and the dotted line shown.

The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 :-Which of the following formulae are correct for describing the motion of the particle over the time-interval t_1 to t_2 :- x(t_2)=x(t_1)+v(t_1)(t_2-t_1)+(1/2)a(t_2-t_1)^2

The tangents to the parabola y^2=4ax at P(at_1^2,2at_1) , and Q(at_2^2,2at_2) , intersect at R. Prove that the area of the triangle PQR is 1/2a^2(t_1-t_2)^3

The vertex of the parabola whose parametric equation is x=t^(2)-t+1,y=t^(2)+t+1,tinR , is

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/(t_2)^ 2))dot