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

(i) Tangents are drawn from the point (alpha, beta) to the parabola y^2 = 4ax . Show that the length of their chord of contact is : 1/|a| sqrt((beta^2 - 4aalpha) (beta^2 + 4a^2)) . Also show that the area of the triangle formed by the tangents from (alpha, beta) to parabola y^2 = 4ax and the chord of contact is (beta^2 - 4aalpha)^(3/2)/(2a) . (ii) Prove that the area of the triangle formed by the tangents at points t_1 and t_2 on the parabola y^2 = 4ax with the chord joining these two points is a^2/2 |t_1 - t_2|^3 .

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

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

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 bounded by the hyperbola and the lines joining its centre to the points corresponding to t_1a n d-t_1 is t_1dot

Prove that the length of the intercept on the normal at the point P(a t^2,2a t) of the parabola y^2=4a x made by the circle described on the line joining the focus and P as diameter is asqrt(1+t^2) .

Prove that the length of the intercept on the normal at the point P(a t^2,2a t) of the parabola y^2=4a x made by the circle described on the line joining the focus and P as diameter is asqrt(1+t^2) .

Write the area of the triangle formed by the coordinate axes and the line (s e ctheta-t a ntheta)x+(s e ctheta+t a ntheta)y=2.

The area of triangle formed by tangents at the parametrie points t_(1),t_(2) and t_(3) , on y^(2) = 4ax is k |(t_(1)-t_(2)) (t_(2)-t_(1)) (t_(3)-t_(1))| then K =

Area of the triangle formed by the threepoints 't_1'. 't_2' and 't_3' on y^2=4ax is K|(t_1-t_2) (t_2-t_3)(t_3-t_1)| then K=

Find the equation of normal to the curves x=t^2, y=2t+1 at point