If the tangent at `(x_1,y_1)` to the curve `x^3+y^3=a^3` meets the curve again in `(x_2,y_2),` then prove that `(x_2)/(x_1)+(y_2)/(y_1)=-1`

If the tangent at (1,1) on y^2=x(2-x)^2 meets the curve again at P , then find coordinates of P

If the join of (x_1,y_1) and (x_2,y_2) makes on obtuse angle at (x_3,y_3), then prove than (x_3-x_1)(x_3-x_2)+(y_3-y_1)(y_3-y_2)<0

The normal at the point (1,1) on the curve 2y + x^(2) = 3 is

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

Find the slope of the tangent to the curve y = x^(3) – x at x = 2.

Find the equations of tangent and normal to the curve y=x^(2)+3x-2 at the point (1, 2). .

The line y = x + 1 is a tangent to the curve y^(2) = 4x at the poin

A triangle has vertices A_i(x_i , y_i)fori=1,2,3 If the orthocentre of triangle is (0,0), then prove that |x_2-x_3y_2-y_3y_1(y_2-y_3)+x_1(x_2-x_3)x_3-x_1y_2-y_3y_2(y_3-y_1)+x_1(x_3-x_1)x_1-x_2y_2-y_3y_3(y_1-y_2)+x_1(x_1-x_2)|=0

Statement 1: The tangent at x=1 to the curve y=x^3-x^2-x+2 again meets the curve at x=0. Statement 2: When the equation of a tangent is solved with the given curve, repeated roots are obtained at point of tangency.