The curves y=x²-1, y=8x-x²-9 touch each other at the point (2, 3). The equation of the common normal is

If the subnormal of the curve xyⁿ=aⁿ⁺¹ is constant, then the value of n is

The equation of tangent to the curve y=x+9/x+5 so that is passes through the origin is

If θ is the angle between the curves y=sin x, y =cosx at x=π/4 then tan θ=

A curve passes through the point (2, 0) and the slope of the tangent at any point is x²-2x for all values of x. The point of maximum or donation the curve is

The function f(x) = tan x has

f(x)= x-1/x is

The function f(x)=cot^-1 x+x increases in the interval

The point P in the first quadrant of the ellipse x²/8+y²/18=1 so that the area of the triangle formed by the tangent at P and the coordinate axes is least

A rectangular sheet of dimensions 30 cm * 80 cm four equal squares of side x cm are removed at the corners, and the sides are then turned up so as to form an open rectangle box. The value of x, so that the value of the box is the greatest is