Prove that the envelope of all parabolas of which the focus is at the origin and the vertex is on the circle \(x^2+y^2=2ax\) is the straight line \(x=2a\).
Prove that the radius of a curvature at any point of a curve is \(r\frac{dr}{dp}\), where \(r\) is the radius vector and \(p\) the perpendicular from the origin on the tangent at the point. Prove that the perpendicular of greatest length, which can be drawn from the centre of an ellipse on a normal, is equal to the difference between the semi-axes of the ellipse; and that the foot of this perpendicular is the centre of curvature of the point from which the normal is drawn.
Evaluate the integrals \[ \int \sec^4\theta d\theta, \quad \int \tan^{-1}x dx, \quad \int \frac{dx}{(x+1)^2(x^2+1)}, \quad \int_0^{\frac{\pi}{2}} \frac{dx}{5+4\cos x}. \]
Prove that the area of a closed curve is \(\frac{1}{2}\int(xdy-ydx)\) taken round the curve. Shew that the area of a loop of the curve \(a^3y^2=4x^2(a^2-x^2)\) is \(\frac{4}{3}a^2\).