Prove that for the curve `y=be^(x//a)` , the subtangent is of constant length and the sub-normal varies as the square of the ordinate .

A curve passing through (1, 0) is such that the ratio of the square of the intercept cut by any tangent on the y-axis to the Sub-normal is equal to the ratio of the product of the Coordinates of the point of tangency to the product of square of the slope of the tangent and the subtangent at the same point, is given by

If the length of sub-normal is equal to the length of sub-tangent at any point (3,4) on the curve y=f(x) and the tangent at (3,4) to y=f(x) meets the coordinate axes at Aa n dB , then the maximum area of the triangle O A B , where O is origin, is 45/2 (b) 49/2 (c) 25/2 (d) 81/2

Find the family of curves, the subtangent at any point of which is the arithmetic mean of the co-ordinate point of tangency.

Prove that all points on the curve y^(2)=4a[x+a sin((x)/(a))] at which the tangent is parallel to the X - axis lie on the parabola y^(2)=4ax .

Prove that the length of a focal chord of a parabola varies inversely as the square of its distance from the vertex.

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.

Find the equations of the tangent and normal to the given curves at the indicated points : y=x^(3) at (1, 1)

Find the equations of the tangent and normal to the given curves at the indicated points : y=x^(2) at (0, 0)

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Find the curve for which the intercept cut off by any tangent on y-axis is proportional to the square of the ordinate of the point of tangency.