Point of intersection of normal at `P(at_1^2, 2at_1)` and `Q(at_2^2, 2at_2)`

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 extremities of focal chord of a parabola are (at_1^2,2at_1) and (at_2^2,2at_2) , then t_1t_2=-1 .

Find the point of intersection of lines : yt_1 = x+at_1^2 and yt_2 = x+ at_2^2

Find the distance between the points : ( at_1^2 , 2at_1) and (at_2 ^2, 2at_2)

Tangents to the parabola y^2=4ax at P(at_1^2,2at_1)and Q(at_2^2,2at_2) meet at T. If DeltaPTQ is right - angled at T, then 1/(PS)+1/(QS) is equal to (where , S is the focus of the given parabola)

Find the area of the triangle whose vertices are : (at_1^2, 2at_1), (at_2^2 , 2at_2), (at_3^2, 2at_3)

From a point P (h, k), in general, three normals can be drawn to the parabola y^2= 4ax. If t_1, t_2,t_3 are the parameters associated with the feet of these normals, then t_1, t_2, t_3 are the roots of theequation at at^2+(2a-h)t-k=0. Moreover, from the line x = - a, two perpendicular tangents canbe drawn to the parabola. If the tangents at the feet Q(at_1^2, 2at_1) and R(at_1^2, 2at_2) to the parabola meet on the line x = -a, then t_1, t_2 are the roots of the equation

Any ordinate MP of an ellipse meets the auxillary circle in Q. Ptove that the locus of the point of intersection of the normals at P and Q is the circle x^(2)+y^(2)=(a+b)^(2) .

Find the equation of the straight line whichpasses throought the two points : (at_1^2 , 2at_1), (at_2^2, 2at_2)