If the normal at `(at_(1)^(2), 2at_(1))" to " y^(2) =4ax` intersect the parabola at `(at_(2)^(2), 2at_(2))` , prove that,
`t_(1)+t_(2)+(2)/(t_1)=0(t_(1) ne 0)`

If the normal at (at_(1)^(2),2at_(1)) "to" y^(2)=4ax intesect the parabola at (at_(2)^(2),2at_(2)) "then" t_(1)+t_(2)+(k)/(t_(1))=0 (t_(1) ne 0) , find k.

If a straight line pasing through the focus of the parabola y^(2) = 4ax intersects the parabola at the points (x_(1), y_(1)) and (x_(2) , y_(2)) then prove that y_(1)y_(2)+4x_(1)x_(2)=0 .

If the normals at P(t_(1))andQ(t_(2)) on the parabola meet on the same parabola, then (A) t_(1)t_(2)=-1 (B) t_(2)=-t_(1)-(2)/(t_(1)) (C) t_(1)t_(2)=1 (D) t_(1)t_(2)=2

The distance between the points (at_1^(2) , 2at_(1)) and (at_(2)^(2) ,2at_(2)) is

If the extremities of a focal chord of the parabola y^(2) = 4ax be (at_(1)^(2) , 2at_(1)) and (at_(2)^(2) ,2at_(2)) , show that t_(1)t_(2) = - 1

If the normal to the parabola y^2=4a x at point t_1 cuts the parabola again at point t_2 , then prove that t_2^2 geq8.

Prove that the normal at (am^(2),2am) to the parabola y^(2)=4ax meets the curve again at an angle tan^(-1)((1)/(2)m) .

Find the area of the triangle with verrtices (at_(1)^(2),2at_(1)),(at_(2)^(2),2at_(2))and(at_(3)^(2),2at_(3))

The normal to the parabola y^(2)=4ax at P(am_(1)^(2),2am_(1)) intersects it again at Q(am_(2)^(2), 2am_(2)) .If A be the vertex of the parabola then show that the area of the triangle APQ is (2a^(2))/(m_(1))(1+m_(1)^(2))(2+m_(1)^(2)) .

If x=tan t and y=tan pt , prove that, (1+x^(2))y_(2)+2(x-py)y_(1)=0