Find the equation of the line joining two points on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), whose eccentric angles are \(\phi_1\) and \(\phi_2\); hence deduce the equation of the tangent at a point. If \(\phi_1, \phi_2, \phi_3, \phi_4\) are the eccentric angles of the points in which the ellipse is cut by a circle, prove that \(\phi_1+\phi_2+\phi_3+\phi_4=2n\pi\). Shew that the circle of curvature at a point whose ordinate is \(\frac{1}{2}b\) passes through an extremity of the minor axis of the ellipse.
If the equations of three straight lines are expressed in the forms \(\alpha=0, \beta=0, \gamma=0\), interpret the equation \(\alpha\beta=\gamma^2\), and shew that \(\alpha+2\lambda\gamma+\lambda^2\beta=0\) is a tangent to the curve. Shew that the equation of the tangents from the origin to the conic \[ (x-a)(y-a) = (x+y-b)^2 \] is \[ a^2(x-y)^2+4\{ax+(a-b)y\}\{(a-b)x+ay\}=0. \]
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.