If the tangent at `(x_(0),y_(0))` to the curve `x^(3)+y^(3)=a^(3)` meet the curve again at `(x_(1)y_(1))` prove that `(x_(1))/(x_(0))+(y_(1))/(y_(0))=-1`

At the point (x_(1),y_(1)) on the curve x^(3)+y^(3)=3axy show that the tangent is (x_(1)^(2)-ay_(1))x+(y_(1)^(2)-ax_(1))y=ax_(1)y_(1)

If the tangent at (1,1) on y^2=x(2-x)^2 meets the curve again atpis

Prove that the area of the triangle formed by the tangents at (x_(1),y_(1)),(x_(2)) "and" (x_(3),y_(3)) to the parabola y^(2)=4ax(agt0) is (1)/(16a)|(y_(1)-y_(2))(y_(2)-y_(3))(y_(3)-y_(1))| sq.units.

If the hyperbola xy=c^(2) intersects the circle x^(2)+y^(2)=a^(2)" is four points "P(x_(1),y_(1)), Q(x_(2),y_(2)), R(x_(3),y_(3)) and S(x_(4),y_(4)) then show that (i) x_(1)+x_(2)+x_(3)+x_(4)=0 (ii) y_(1)+y_(2)+y_(3)+y_(4)=0 (iii) x_(1)x_(2)x_(3)x_(4)=c^(4) (iv) y_(1)y_(2)y_(3)y_(4)=c^(4)

The equation of the tangent to the curve y(x-2)(x-3)-x+7=0 where the curve cuts x-axis is

Tangent at any point of the curve (x/a)^(2//3)+(y/b)^(2//3)=1 makes intercepts x_1 and y_1 on the axes. Then

The tangent to the circle x^(2)+y^(2)-4x+2y+k=0 at (1,1) is x-2y+1=0 then k=

If the normals at (x_(1),y_(1)) and (x_(2) ,y_(2)) on y^(2) = 4ax meet again on parabola then x_(1)x_(2)+y_(1)y_(2)=