The parabola `x = y^2 + ay+b` intersect the parabola `x^2=y` at (1,1) at right .which of the foolowing is /are correct?

x+y=2 and x-y= 2 are tangents on a parabola at (1,1) and (4,2) respectivley. Which of the followings is/are correct.

The line y=(2x+a) will not intersect the parabola y^(2)=2x if

If a straight line passing through the focus of the parabola x ^(2) =4ay intersects the parabola at the points (x_(1) , y_(1)) and (x_(2), y_(2)) then the value of x_(1)x_(2) is-

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 a straight line passing through the focus of the parabola y^(2) = 4ax intersectts the parabola at the points (x_(1), y_(1)) and (x_(2), y_(2)) , then prove that x_(1)x_(2)=a^(2) .

if the tangent to the parabola y=x(2-x) at the point (1,1) intersects the parabola at P. find the co-ordinate of P.

The line y=x-2 cuts the parabola y^(2)=8x in the points A and B. The normals drawn to the parabola at A and B intersect at G. A line passing through G intersects the parabola at right angles at the point C,and the tangents at A and B intersect at point T.

The equation of a parabola is (x + y - 2)^2 = sqrt2 (x - y) then (a) Vertex of the parabola is (1,1) (b) latus rectum of parabola is 2. (c) The equation of tangent at vertex is x+y-2=0. (d) the perpendicular from focus on any tangent of the parabola is x-y=0.