Find The Locus Of A Point That Divides A Chord Of Slope 2 Of The Parabola y2= 4x internally In The Ratio 1: 2.

The point of intersection of the tangents drawn to the curve x²y = 1 – y at the points where it is meet by the curve xy = 1 – y is given by

The slope of the normal at the point with abscissa x = –2 of the graph of the function f (x) = | x² – x | is

The tangent to the graph of the function y = f (x) at the point with abscissax = 1 form an angle of π/6 and at the point x = 2 an angle of π/3 and at the point x = 3 an angle of π/4. The value of 

The equations of the tangents to the curve y = x⁴ from the point (2, 0) not on the curve, are given by

The value of parameter a so that line (3 – a) x + ay + (a²ⁿ – 1) = 0 is normal to the curve xy = 1, may lie in the interval

A cylindrical gas container is closed at the top and open at the bottom; if the iron plate forming the cylindrical sides. The ratio of the height to diameter of the diameter of the cylindrical using minimum material for the same capacity is

The critical points of the function f (x) where 

Find the abscissa of the point on the curve ay² = x³, the normal at which cuts of equal intercept from the axes.

Find the condition that the curves; ax² + by² = 1 and a ‘x² + b’ y² = 1 may cut each other orthogonally (at right angles).

Find the acute angle between the curves 

 at their points of intersection when x > 0.