The coordinates of end points of a focal chord of an ellipse are `(x_(1),y_(1))and(x_(2),x_(2))` Prove that `y_(1)y_(2)+4x_(1)x_(2)=0`.

The co-ordinates of end points of a focal chord of an ellipse are (x_1,y_1) and (x_2,y_2) .Prove that y_1y_2+4x_1x_2=0

The coordinates of the ends of a focal chord of the parabola y^2=4a x are (x_1, y_1) and (x_2, y_2) . Then find the value of x_1x_2+y_1y_2 .

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 .

The coordinates of any point on the line-segment joining the points ( x _(1) ,y_(1) , z_(1)) and (x_(2), y_(2),z_(2)) are ((x_(1)+kx_(2))/(k + 1),(y _(1) + ky_(2))/(k + 1),(z_(1)+kz_(2))/(k + 1)) , then the value of k will be _

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x_1,y_1) and (x_2,y_2), respectively, then (a) x_1=y_2 (b) x_1=y_1 (c) y_1=y_2 (d) x_2=y_1

Find the coordinates of the centroid fof the triangle whose vertices are (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)) and (x_(3),y_(3),z_(3))

If y=log (1+ sin x) , prove that y_(4)+y_(3)y_(1)+y_(2)^(2)=0 .

If O be the origin and if coordinates of any two points Q_(1) and Q_(2) be (x_(1),y_(1)) and (x_(2),y_(2)) respectively, prove that overline(OQ_(1))* overline(OQ_(2)) cos angle Q_(1)OQ_(2)=x_(1)x_(2)+y_(1)+y_(2) .

The locus of the middle points of the focal chords of the parabola, y^2=4x is:

If y=x log((1)/(ax)+(1)/(a)) , prove that, x(x+1)y_(2)+xy_(1)=y-1 .