Differentiate \(\sin^{-1}\frac{a+b\cos x}{b+a\cos x}\). If \(\log x + \log y = \frac{x}{y}\), prove that \(\frac{dy}{dx} = \frac{y(x-y)}{x(x+y)}\).
If \(y=\sin(a\sin^{-1}x)\), prove that \((1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}+a^2y=0\). Hence or otherwise prove that \[ \sin(a\sin^{-1}x) = ax - \frac{a(a^2-1)}{3!}x^3 + \frac{a(a^2-1)(a^2-9)}{5!}x^5 - \dots. \]
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 \(\frac{4}{3}a^2\).
Prove that, if the middle points of the coplanar lines \(AB, BC, CD, DA\) are concyclic, \(AC\) is at right angles to \(BD\): deduce that, if the middle points of five of the joins of four points are on a circle, so also is the middle point of the sixth join.
Prove that in successive inversion with regard to two orthogonal circles the order of inversion is immaterial: shew also that, if \(P\) be a point, \(P_1\) and \(P_2\) its inverse points with regard to any two circles not orthogonal, and \(P_{12}, P_{21}\) their inverses, the five points are on a circle cutting the two circles orthogonally.
The lines \(AP, BP\) through fixed points \(A\) and \(B\) are such that the angles made with the line from \(A\) to \(B\) have a constant sum; shew that the locus of \(P\) is a rectangular hyperbola of which \(AB\) is a diameter. Deduce that the points of contact of tangents in a given direction to confocal conics lie on a rectangular hyperbola through the foci.
Two fixed lines which do not intersect are taken in space: shew that in a definite direction one and only one line can in general be drawn to intersect both lines. What are the exceptional directions? Prove also that, if directions be chosen parallel to a fixed plane, the corresponding lines have intercepts between the two fixed lines such that their middle points lie on a line.