Question

The Centre of curvature of the parabola x = at2 , y = 2at at the point “t” is

Answer 1

Answer:

If S = 0 be the equation to a conic T = 0 the equation to the tangent at the point P whose coordinates are at2 and 2at and L = 0 the equation to any straight line passing through P we know thatis the equation to the conic section passing through three coincident points at P and through the other point in which L = 0 meets S = 0.If λand L be so chosen that this conic is a circle it will be the circle of curvature at P and by the last article we know that L = 0 will be equally inclined to the axis with T = 0.In the case of a parabolaAlso the equation to a line through at2 2at equally inclined with T = 0 to the axis isThe equation to the circle of curvature is thereforeOn substituting this value of λ we have as the required equation The circle of curvature has therefore its centre at the point 2a + 3at2 - 2at3 and its radius equal to