Show that the area formed by the normals to `y^2=4ax` at the points `t_1,t_2,t_3` is

Find the equation of the tangent and normal to the parabola y^2=4a x at the point (a t^2,\ 2a t) .

Show that the normal to the rectangular hyperbola xy = c^(2) at the point t meets the curve again at a point t' such that t^(3)t' = - 1 .

Show that the tangents and the normals to the parabola x=at^(2),y=2at at points corresponding to t=1 and t=-1 form a square.

For any real t,x=(1)/(2)(e^(t)+e^(-t)),y=(1)/(2)(e^(t)-e^(-t)) is a point on the hyperbola x^(2)-y^(2)=1 show that the area bouyped by the hyperbola and the lines joining its centre to the points corresponding to t_(1)and-t_(1) ist _(1) .

If 't_1' and 't_2' be the ends of a focal chord of the parabola y^2=4ax then t_1t_2 is equal to

If the orthocentre of the triangle formed by the points t_1,t_2,t_3 on the parabola y^2=4ax is the focus, the value of |t_1t_2+t_2t_3+t_3t_1| is

If the tangent to y^2=4ax at the point (at^2,2at) , where |t|gt1 is a normal to x^2-y^2=a^2 at the point (asectheta, a tan theta) , then

Normals are drawn to the parabola y^(2)=4ax at the points A,B,C whose parameters are t_(1),t_(2) and t_(3) ,respectively.If these normals enclose a triangle PQR,then prove that its area is (a^(2))/(2)(t-t_(2))(t_(2)-t_(3))(t_(3)-t_(1))(t_(1)+t_(2)+t_(3))^(2) Also prove that Delta PQR=Delta ABC(t_(1)+t_(2)+t_(3))^(2)

Show that the locus represented by x = 1/2 a (t + 1/t) , y = 1/2 a (t - 1/t) is a rectangular hyperbola. Show also that equation to the normal at the point 't' is x/(t^(2) + 1) + y/(t^(2) - 1) = a/t .

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