Tangents are drawn from the origin to curve `y=sinxdot` Prove that points of contact lie on `y^2=(x^2)/(1+x^2)`

Three normals are drawn from the point (14,7) to the curve y^2-16x-8y=0 . Find the coordinates of the feet of the normals.

Let f(x,y) be a curve in the x-y plane having the property that distance from the origin of any tangent to the curve is equal to distance of point of contact from the y-axis. It f(1,2)=0, then all such possible curves are

Tangent is drawn at any point (x_1,y_1) on the parabola y^2=4ax . Now tangents are drawn from any point on this tangent to the circle x^2+y^2=a^2 such that all the chords of contact pass throught a fixed point (x_2,y_2) Prove that 4(x_1/x_2)+(y_1/y_2)^2=0 .

Find the slope of the tangent to curve y=x^(3)-x+1 at the point whose x- coordinate is 2.

Statement I Tangent drawn at the point (0, 1) to the curve y=x^(3)-3x+1 meets the curve thrice at one point only. statement II Tangent drawn at the point (1, -1) to the curve y=x^(3)-3x+1 meets the curve at one point only.

Tangents are drawn from the point (-1, 2) to the parabola y^2 =4x The area of the triangle for tangents and their chord of contact is

y=3x is tangent to the parabola 2y=ax^2+ab . If b=36, then the point of contact is

The x-intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of the curve through the point (1,1) is

The equation of the tangents to the curve (1+x^(2))y=1 at the points of its intersection with the curve (x+1)y=1 , is given by

Show that the angle between the tangent at any point P and the line joining P to the origin O is same at all points on the curve log(x^2+y^2)=ktan^(-1)(y/x)