Let `y=f(x)` be a curve passing through `(1,1)` such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Form the differential equation and determine all such possible curves.

The area of the triangle formed with coordinate axes and the tangent at (x_(1),y_(1)) on the circle x^(2)+y^(2)=a^(2) is

The area bounded by the coordinate axes and the curve sqrt(x) + sqrt(y) = 1 , is

Form a differential equation of the family of the curves y^2 =4ax .

The slope of the tangent to the curve y=x^(2) -x at the point where the line y = 2 cuts the curve in the first quadrant, is

Form a differential equation of the family of the curves y^(2)=4ax

If the area of the triangle included between the axes and any tangent to the curve x^n y=a^n is constant, then find the value of ndot

If the area of the triangle included between the axes and any tangent to the curve x^n y=a^n is constant, then find the value of ndot

A curve y=f(x) passes through 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, then the equation of the curve is

Let y=f(x)be curve passing through (1,sqrt3) such that tangent at any point P on the curve lying in the first quadrant has positive slope and the tangent and the normal at the point P cut the x-axis at A and B respectively so that the mid-point of AB is origin. Find the differential equation of the curve and hence determine f(x).

Let C be a curve passing through M(2,2) such that the slope of the tangent at any point to the curve is reciprocal of the ordinate of the point. If the area bounded by curve C and line x=2 is A, then the value of (3A)/(2) is__.