If P and Q are two points whose coordinates are `(a t^2,2a t)a n d(a/(t^2),(2a)/t)` respectively and S is the point (a,0). Show that `1/(S P)+1/(s Q)` is independent of t.

Find the locus of a point whose coordinate are given by x = t+t^(2), y = 2t+1 , where t is variable.

If the coordinates of a vartiable point P be (t+(1)/(t), t-(1)/(t)) , where t is the variable quantity, then the locus of P is

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

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 all the points of discontinuity of the function f(t) = (1/(t^2 + t-2)) , where t = 1/(x-1)

Let a, r, s, t be non-zero real numbers. Let P(at^(2),2at),Q(ar^(2),2ar)andS(as^(2),2as) be distinct points on the parabola y^(2)=4ax . Suppose that PQ is the focal chord and lines QR and PK are parallel, where K the point (2a,0). If st=1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

Statement 1 : If a circle S=0 intersects a hyperbola x y=4 at four points, three of them being (2, 2), (4, 1) and (6,2/3), then the coordinates of the fourth point are (1/4,16) . Statement 2 : If a circle S=0 intersects a hyperbola x y=c^2 at t_1,t_2,t_3, and t_4 then t_1t_2t_3t_4=1

If t_1 a n d t_2 are roots of eth equation t^2+lambdat+1=0, where lambda is an arbitrary constant. Then prove that the line joining the points (a t1^2,2a t_1)a d n(a t2^2,2a t_2) always passes through a fixed point. Also, find the point.

If (7 + 4 sqrt(3))^(n) = s + t , where n and s are positive integers and t is a proper fraction , show that (1 - t) (s + t) = 1