For the curve `y=a1n(x^2-a^2)` , show that the sum of length of tangent and sub-tangent at any point is proportional to product of coordinates of point of tangency.

The equation of the curve which is such that the portion of the axis of x cut off between the origin and tangent at any point is proportional to the ordinate of that point is

A curve y=f(x) passes through the point P(1,1). The normal to the curve at P is a(y-1)+(x-1)=0. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, find the equation of the curve.

Show that a triangle made by a tangent at any point on the curve xy =c^(2) and the coordinates axes is of constant area.

Find the equation of the curve whose length of the tangent at any point on ot, intercepted between the coordinate axes is bisected by the point of contact.

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

For the curve y=f(x) prove that (length of normal)^2/(length of tangent)^2 =length of sub-normal/length of sub-tangent

For the curve x y=c , prove that the portion of the tangent intercepted between the coordinate axes is bisected at the point of contact.

Show that for the curve b y^2=(x+a)^3, the square of the sub-tangent varies as the sub-normal.

Find the point on the curve y=x^(3) where the slop of the tangent is equal to x-coordinate of the point.

Find the equation of a curve passing through the point (0,1).If the slope of the tangent to the curve at any point (x,y) is equal to the sum of the x coordinate(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.